| L(s) = 1 | + 8·2-s + 40·4-s + 56·7-s + 160·8-s + 5·9-s − 49·11-s + 52·13-s + 448·14-s + 560·16-s + 27·17-s + 40·18-s + 51·19-s − 392·22-s − 9·23-s + 416·26-s − 48·27-s + 2.24e3·28-s − 161·29-s + 38·31-s + 1.79e3·32-s + 216·34-s + 200·36-s + 16·37-s + 408·38-s + 547·41-s + 609·43-s − 1.96e3·44-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s + 3.02·7-s + 7.07·8-s + 5/27·9-s − 1.34·11-s + 1.10·13-s + 8.55·14-s + 35/4·16-s + 0.385·17-s + 0.523·18-s + 0.615·19-s − 3.79·22-s − 0.0815·23-s + 3.13·26-s − 0.342·27-s + 15.1·28-s − 1.03·29-s + 0.220·31-s + 9.89·32-s + 1.08·34-s + 0.925·36-s + 0.0710·37-s + 1.74·38-s + 2.08·41-s + 2.15·43-s − 6.71·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(128.1454744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(128.1454744\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - p T )^{4} \) |
| 5 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - p T )^{4} \) |
| good | 3 | $C_2 \wr S_4$ | \( 1 - 5 T^{2} + 16 p T^{3} - 128 T^{4} + 16 p^{4} T^{5} - 5 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 8 p T + 2127 T^{2} - 56354 T^{3} + 1179224 T^{4} - 56354 p^{3} T^{5} + 2127 p^{6} T^{6} - 8 p^{10} T^{7} + p^{12} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 49 T + 3391 T^{2} + 138994 T^{3} + 5643870 T^{4} + 138994 p^{3} T^{5} + 3391 p^{6} T^{6} + 49 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 27 T + 2197 T^{2} - 22226 T^{3} + 35190538 T^{4} - 22226 p^{3} T^{5} + 2197 p^{6} T^{6} - 27 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 51 T + 56 p^{2} T^{2} - 1144575 T^{3} + 184346126 T^{4} - 1144575 p^{3} T^{5} + 56 p^{8} T^{6} - 51 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 9 T + 19148 T^{2} - 1050327 T^{3} + 211262118 T^{4} - 1050327 p^{3} T^{5} + 19148 p^{6} T^{6} + 9 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 161 T + 62851 T^{2} + 8580410 T^{3} + 2244057936 T^{4} + 8580410 p^{3} T^{5} + 62851 p^{6} T^{6} + 161 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 38 T + 82751 T^{2} - 2524314 T^{3} + 3420131180 T^{4} - 2524314 p^{3} T^{5} + 82751 p^{6} T^{6} - 38 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 16 T + 92424 T^{2} + 3890912 T^{3} + 4714532798 T^{4} + 3890912 p^{3} T^{5} + 92424 p^{6} T^{6} - 16 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 547 T + 282444 T^{2} - 78101601 T^{3} + 24924328566 T^{4} - 78101601 p^{3} T^{5} + 282444 p^{6} T^{6} - 547 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 609 T + 210408 T^{2} - 27097781 T^{3} + 3443560014 T^{4} - 27097781 p^{3} T^{5} + 210408 p^{6} T^{6} - 609 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 254 T + 329955 T^{2} + 78532974 T^{3} + 46994253256 T^{4} + 78532974 p^{3} T^{5} + 329955 p^{6} T^{6} + 254 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 961 T + 826473 T^{2} - 439333422 T^{3} + 201643806858 T^{4} - 439333422 p^{3} T^{5} + 826473 p^{6} T^{6} - 961 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 230 T + 570243 T^{2} + 184187238 T^{3} + 147920246352 T^{4} + 184187238 p^{3} T^{5} + 570243 p^{6} T^{6} + 230 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 1393 T + 1287731 T^{2} - 871191462 T^{3} + 481233034676 T^{4} - 871191462 p^{3} T^{5} + 1287731 p^{6} T^{6} - 1393 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 579 T + 478955 T^{2} - 66957006 T^{3} + 73599298802 T^{4} - 66957006 p^{3} T^{5} + 478955 p^{6} T^{6} - 579 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 680 T + 1526320 T^{2} - 718747160 T^{3} + 834788988798 T^{4} - 718747160 p^{3} T^{5} + 1526320 p^{6} T^{6} - 680 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 1321 T + 1890170 T^{2} - 1517221359 T^{3} + 1160323729322 T^{4} - 1517221359 p^{3} T^{5} + 1890170 p^{6} T^{6} - 1321 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 55 T + 544628 T^{2} - 9199713 T^{3} + 329780786422 T^{4} - 9199713 p^{3} T^{5} + 544628 p^{6} T^{6} + 55 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 2733 T + 4979865 T^{2} - 5812678818 T^{3} + 5207004649132 T^{4} - 5812678818 p^{3} T^{5} + 4979865 p^{6} T^{6} - 2733 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 1233 T + 2665102 T^{2} + 2131465291 T^{3} + 2654755754554 T^{4} + 2131465291 p^{3} T^{5} + 2665102 p^{6} T^{6} + 1233 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 1250 T + 2495976 T^{2} - 1844678942 T^{3} + 2573508001070 T^{4} - 1844678942 p^{3} T^{5} + 2495976 p^{6} T^{6} - 1250 p^{9} T^{7} + p^{12} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.17974575182505976891922135176, −6.82030525931740493629307419425, −6.49500516102182579860576941327, −6.44514597265337479662479540840, −6.01663987694777599169674614651, −5.68531253569788687703482767345, −5.63790132492773667237855687565, −5.34631264018484191107742989911, −5.24181917168181349055555506252, −4.92368756401968252689892952692, −4.88549531679258987808338625536, −4.41379863714528606624197143969, −4.36595360976470844219781337698, −3.99830065172604267297895393791, −3.62918893561899683872168337741, −3.53351675285351490675953357109, −3.42830441525640553306252229339, −2.66335058184802715694427739823, −2.39552003818372709361648200977, −2.25912339245938537004282696603, −2.10490838248663253492119317506, −1.71305215942496138245145684718, −1.17554651769401414081471723671, −0.883822737783078875389563975974, −0.73932665550596750286870195681,
0.73932665550596750286870195681, 0.883822737783078875389563975974, 1.17554651769401414081471723671, 1.71305215942496138245145684718, 2.10490838248663253492119317506, 2.25912339245938537004282696603, 2.39552003818372709361648200977, 2.66335058184802715694427739823, 3.42830441525640553306252229339, 3.53351675285351490675953357109, 3.62918893561899683872168337741, 3.99830065172604267297895393791, 4.36595360976470844219781337698, 4.41379863714528606624197143969, 4.88549531679258987808338625536, 4.92368756401968252689892952692, 5.24181917168181349055555506252, 5.34631264018484191107742989911, 5.63790132492773667237855687565, 5.68531253569788687703482767345, 6.01663987694777599169674614651, 6.44514597265337479662479540840, 6.49500516102182579860576941327, 6.82030525931740493629307419425, 7.17974575182505976891922135176
