| L(s) = 1 | + 64·2-s + 3.07e3·4-s + 6.25e3·5-s + 1.96e4·7-s + 1.31e5·8-s + 4.00e5·10-s + 5.88e5·11-s + 4.37e5·13-s + 1.25e6·14-s + 5.24e6·16-s + 9.59e5·17-s + 1.84e6·19-s + 1.92e7·20-s + 3.76e7·22-s + 2.51e7·23-s + 2.92e7·25-s + 2.79e7·26-s + 6.02e7·28-s + 2.83e7·29-s + 1.29e8·31-s + 2.01e8·32-s + 6.13e7·34-s + 1.22e8·35-s − 2.71e8·37-s + 1.18e8·38-s + 8.19e8·40-s + 8.19e8·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.894·5-s + 0.440·7-s + 1.41·8-s + 1.26·10-s + 1.10·11-s + 0.326·13-s + 0.623·14-s + 5/4·16-s + 0.163·17-s + 0.170·19-s + 1.34·20-s + 1.55·22-s + 0.814·23-s + 3/5·25-s + 0.462·26-s + 0.661·28-s + 0.256·29-s + 0.809·31-s + 1.06·32-s + 0.231·34-s + 0.394·35-s − 0.643·37-s + 0.241·38-s + 1.26·40-s + 1.10·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(18.12419620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(18.12419620\) |
| \(L(\frac{13}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{5} T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{5} T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 400 p^{2} T + 405495270 p T^{2} - 400 p^{13} T^{3} + p^{22} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 588120 T + 219480823666 T^{2} - 588120 p^{11} T^{3} + p^{22} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 437440 T + 3045441554010 T^{2} - 437440 p^{11} T^{3} + p^{22} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 959244 T + 48287200405750 T^{2} - 959244 p^{11} T^{3} + p^{22} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 97072 p T + 169704154197894 T^{2} - 97072 p^{12} T^{3} + p^{22} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 25155840 T + 899601164555854 T^{2} - 25155840 p^{11} T^{3} + p^{22} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 28343700 T + 24555291966922222 T^{2} - 28343700 p^{11} T^{3} + p^{22} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 129007432 T + 2492635579563918 T^{2} - 129007432 p^{11} T^{3} + p^{22} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 271351400 T + 77189780505156090 T^{2} + 271351400 p^{11} T^{3} + p^{22} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 819882960 T + 830174502197488882 T^{2} - 819882960 p^{11} T^{3} + p^{22} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 363579680 T + 1778889785883320358 T^{2} + 363579680 p^{11} T^{3} + p^{22} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 1860270000 T + 5168201679109024606 T^{2} - 1860270000 p^{11} T^{3} + p^{22} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5146147236 T + 22370462635774079518 T^{2} - 5146147236 p^{11} T^{3} + p^{22} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8007070440 T + 76183881567836967922 T^{2} - 8007070440 p^{11} T^{3} + p^{22} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 9401147524 T + \)\(10\!\cdots\!66\)\( T^{2} - 9401147524 p^{11} T^{3} + p^{22} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 28533673600 T + \)\(44\!\cdots\!82\)\( T^{2} - 28533673600 p^{11} T^{3} + p^{22} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12025917360 T + \)\(24\!\cdots\!78\)\( T^{2} + 12025917360 p^{11} T^{3} + p^{22} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 35386174300 T + \)\(87\!\cdots\!54\)\( T^{2} - 35386174300 p^{11} T^{3} + p^{22} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 50073898888 T + \)\(16\!\cdots\!94\)\( T^{2} - 50073898888 p^{11} T^{3} + p^{22} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 43521426432 T + \)\(22\!\cdots\!90\)\( T^{2} + 43521426432 p^{11} T^{3} + p^{22} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 17035946280 T + \)\(24\!\cdots\!94\)\( T^{2} - 17035946280 p^{11} T^{3} + p^{22} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 156152158420 T + \)\(20\!\cdots\!90\)\( T^{2} - 156152158420 p^{11} T^{3} + p^{22} T^{4} \) |
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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
−12.05413962867982538492896637226, −11.90985739390357505936111835897, −10.99442484251961436053006101109, −10.98819817895143394328337870375, −9.927194443420091592927134715113, −9.711372194126064267643924645651, −8.737038994864635852831789470341, −8.387085255794976106210115726753, −7.37503937113244035233563242255, −6.88585510373137910301578789652, −6.33126720336790119318030925489, −5.85348680346513920663327689986, −5.07243549487340198761452389970, −4.84516713740636075730460905788, −3.69296980672682485247182792433, −3.68487196096754601063081870501, −2.37505811811780741075051445416, −2.27536258003813227176570777807, −1.08769096542112507883739082542, −0.982007450096742480190873144553,
0.982007450096742480190873144553, 1.08769096542112507883739082542, 2.27536258003813227176570777807, 2.37505811811780741075051445416, 3.68487196096754601063081870501, 3.69296980672682485247182792433, 4.84516713740636075730460905788, 5.07243549487340198761452389970, 5.85348680346513920663327689986, 6.33126720336790119318030925489, 6.88585510373137910301578789652, 7.37503937113244035233563242255, 8.387085255794976106210115726753, 8.737038994864635852831789470341, 9.711372194126064267643924645651, 9.927194443420091592927134715113, 10.98819817895143394328337870375, 10.99442484251961436053006101109, 11.90985739390357505936111835897, 12.05413962867982538492896637226
