| L(s) = 1 | + 14·3-s − 42·5-s + 181·9-s + 294·11-s − 140·13-s − 588·15-s + 1.30e3·17-s + 1.44e3·19-s − 2.64e3·23-s − 4.92e3·25-s + 5.72e3·27-s + 1.66e3·29-s + 1.47e4·31-s + 4.11e3·33-s + 5.18e3·37-s − 1.96e3·39-s − 5.12e3·41-s + 4.52e3·43-s − 7.60e3·45-s + 1.49e4·47-s + 1.82e4·51-s + 2.40e4·53-s − 1.23e4·55-s + 2.01e4·57-s − 3.88e4·59-s − 2.36e4·61-s + 5.88e3·65-s + ⋯ |
| L(s) = 1 | + 0.898·3-s − 0.751·5-s + 0.744·9-s + 0.732·11-s − 0.229·13-s − 0.674·15-s + 1.09·17-s + 0.916·19-s − 1.04·23-s − 1.57·25-s + 1.51·27-s + 0.368·29-s + 2.76·31-s + 0.657·33-s + 0.622·37-s − 0.206·39-s − 0.476·41-s + 0.372·43-s − 0.559·45-s + 0.990·47-s + 0.981·51-s + 1.17·53-s − 0.550·55-s + 0.823·57-s − 1.45·59-s − 0.812·61-s + 0.172·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.817048578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.817048578\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 14 T + 5 p T^{2} - 14 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 21 T + p^{5} T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 - 294 T + 114391 T^{2} - 294 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 140 T + 728766 T^{2} + 140 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 1302 T + 3095035 T^{2} - 1302 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 1442 T + 4119519 T^{2} - 1442 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2646 T + 8890015 T^{2} + 2646 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 1668 T + 40800574 T^{2} - 1668 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 14798 T + 109078503 T^{2} - 14798 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5182 T + 71101515 T^{2} - 5182 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 5124 T + 23950966 T^{2} + 5124 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4520 T + 240418566 T^{2} - 4520 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 14994 T + 427056103 T^{2} - 14994 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 24006 T + 935516275 T^{2} - 24006 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 38850 T + 1324947343 T^{2} + 38850 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 23618 T + 1734277563 T^{2} + 23618 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 32002 T + 1139838495 T^{2} + 32002 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 89376 T + 5425689166 T^{2} - 89376 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 47138 T + 4432072947 T^{2} + 47138 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 40970 T + 6023150703 T^{2} - 40970 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 68376 T + 8908447510 T^{2} + 68376 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 123102 T + 13551221779 T^{2} - 123102 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 43652 T + 8867162790 T^{2} + 43652 p^{5} T^{3} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.781033030044834136119114271841, −9.386328555661137466794587455250, −8.660911861215947246859385904483, −8.629000350234050557449025953754, −7.80542821429889573465511720678, −7.76291644811865227322924008998, −7.49475661433990486094304128338, −6.76167899013312170564189480595, −6.19480858536123440781376662154, −6.03557594406199370798544250606, −5.20324449079067706237171438206, −4.68266917979253238955802365286, −4.15584153152647268606535579632, −3.90839344006863622562427890211, −3.21741337691319608164364491180, −2.90364524360692666053296665477, −2.25744959001926327203080281883, −1.58262588482208672471368590753, −0.927685826378288273855472556890, −0.56260882415400124601759883494,
0.56260882415400124601759883494, 0.927685826378288273855472556890, 1.58262588482208672471368590753, 2.25744959001926327203080281883, 2.90364524360692666053296665477, 3.21741337691319608164364491180, 3.90839344006863622562427890211, 4.15584153152647268606535579632, 4.68266917979253238955802365286, 5.20324449079067706237171438206, 6.03557594406199370798544250606, 6.19480858536123440781376662154, 6.76167899013312170564189480595, 7.49475661433990486094304128338, 7.76291644811865227322924008998, 7.80542821429889573465511720678, 8.629000350234050557449025953754, 8.660911861215947246859385904483, 9.386328555661137466794587455250, 9.781033030044834136119114271841
