| L(s) = 1 | − 20·2-s + 144·4-s + 1.22e3·5-s + 2.24e3·8-s + 3.13e3·9-s − 2.44e4·10-s − 1.09e4·13-s − 8.16e4·16-s + 1.46e5·17-s − 6.27e4·18-s + 1.75e5·20-s + 3.35e5·25-s + 2.18e5·26-s − 2.56e5·29-s + 1.05e6·32-s − 2.92e6·34-s + 4.51e5·36-s − 6.94e6·37-s + 2.73e6·40-s + 4.29e6·41-s + 3.82e6·45-s + 9.57e6·49-s − 6.70e6·50-s − 1.57e6·52-s + 1.64e6·53-s + 5.12e6·58-s − 2.94e7·61-s + ⋯ |
| L(s) = 1 | − 5/4·2-s + 9/16·4-s + 1.95·5-s + 0.546·8-s + 0.478·9-s − 2.43·10-s − 0.383·13-s − 1.24·16-s + 1.75·17-s − 0.597·18-s + 1.09·20-s + 0.857·25-s + 0.478·26-s − 0.362·29-s + 1.01·32-s − 2.18·34-s + 0.269·36-s − 3.70·37-s + 1.06·40-s + 1.51·41-s + 0.933·45-s + 1.66·49-s − 1.07·50-s − 0.215·52-s + 0.208·53-s + 0.453·58-s − 2.13·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.9463880618\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9463880618\) |
| \(L(5)\) |
|
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 | $C_2$ | \( 1 + 5 p^{2} T + p^{8} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 1046 p T^{2} + p^{16} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 122 p T + p^{8} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 195362 p^{2} T^{2} + p^{16} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 87015362 T^{2} + p^{16} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 5470 T + p^{8} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 73090 T + p^{8} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 33587484482 T^{2} + p^{16} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 100353384578 T^{2} + p^{16} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 128222 T + p^{8} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1701165473282 T^{2} + p^{16} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 3472030 T + p^{8} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2146882 T + p^{8} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 11766582970942 T^{2} + p^{16} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 10537750788862 T^{2} + p^{16} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 824290 T + p^{8} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 279781405698242 T^{2} + p^{16} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 14746078 T + p^{8} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 579369070794818 T^{2} + p^{16} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 1290076545985922 T^{2} + p^{16} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 5725630 T + p^{8} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 1744457179595522 T^{2} + p^{16} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 1804713576833858 T^{2} + p^{16} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 83324222 T + p^{8} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 120619010 T + p^{8} T^{2} )^{2} \) |
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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
−24.38655660422666686374795816593, −23.05283787075017977522675009153, −22.24342160857562206932762835749, −21.15962427867806757942742231853, −21.04356257984729831813228989909, −19.64969778921455067094912119585, −18.79159807457104404394184152711, −18.15585329877628661872716835672, −17.23027320724048646915557601166, −17.02219167945562673525700051124, −15.77275031872313453029833508855, −14.18255958717057463814967712833, −13.63807897941524328856721670403, −12.33245402130425562647369855681, −10.42394208697097118220815736412, −9.931355639451609166898450403277, −9.012485997895990053334791366615, −7.36865938776434711337216230376, −5.61132311738355574390567068510, −1.67164860332379608344561487750,
1.67164860332379608344561487750, 5.61132311738355574390567068510, 7.36865938776434711337216230376, 9.012485997895990053334791366615, 9.931355639451609166898450403277, 10.42394208697097118220815736412, 12.33245402130425562647369855681, 13.63807897941524328856721670403, 14.18255958717057463814967712833, 15.77275031872313453029833508855, 17.02219167945562673525700051124, 17.23027320724048646915557601166, 18.15585329877628661872716835672, 18.79159807457104404394184152711, 19.64969778921455067094912119585, 21.04356257984729831813228989909, 21.15962427867806757942742231853, 22.24342160857562206932762835749, 23.05283787075017977522675009153, 24.38655660422666686374795816593
