| L(s) = 1 | + 4.68·2-s − 7.83·3-s − 490.·4-s − 625·5-s − 36.6·6-s + 2.40e3·7-s − 4.69e3·8-s − 1.96e4·9-s − 2.92e3·10-s + 6.12e4·11-s + 3.84e3·12-s + 1.27e5·13-s + 1.12e4·14-s + 4.89e3·15-s + 2.28e5·16-s − 3.74e5·17-s − 9.18e4·18-s + 3.51e5·19-s + 3.06e5·20-s − 1.88e4·21-s + 2.86e5·22-s + 1.55e6·23-s + 3.67e4·24-s + 3.90e5·25-s + 5.98e5·26-s + 3.08e5·27-s − 1.17e6·28-s + ⋯ |
| L(s) = 1 | + 0.206·2-s − 0.0558·3-s − 0.957·4-s − 0.447·5-s − 0.0115·6-s + 0.377·7-s − 0.404·8-s − 0.996·9-s − 0.0925·10-s + 1.26·11-s + 0.0534·12-s + 1.24·13-s + 0.0781·14-s + 0.0249·15-s + 0.873·16-s − 1.08·17-s − 0.206·18-s + 0.618·19-s + 0.428·20-s − 0.0211·21-s + 0.261·22-s + 1.16·23-s + 0.0226·24-s + 0.200·25-s + 0.256·26-s + 0.111·27-s − 0.361·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.500455391\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.500455391\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 625T \) |
| 7 | \( 1 - 2.40e3T \) |
| good | 2 | \( 1 - 4.68T + 512T^{2} \) |
| 3 | \( 1 + 7.83T + 1.96e4T^{2} \) |
| 11 | \( 1 - 6.12e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.27e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.74e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 3.51e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.55e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.12e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.54e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 9.25e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 8.05e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.58e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 8.88e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.68e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 8.14e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 2.04e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 8.88e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.24e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.91e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.39e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.05e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.03e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.77e8T + 7.60e17T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.31229933959855414823111856959, −13.54697554992937124912407446891, −12.02648527071599864204168778327, −11.02569139384071029953096227765, −9.115993369649615658482087704142, −8.383427443423370046826559490756, −6.36163194583397904265974623366, −4.76905838600592736703370924403, −3.43165971573998681602855969436, −0.883107660759754934954296433948,
0.883107660759754934954296433948, 3.43165971573998681602855969436, 4.76905838600592736703370924403, 6.36163194583397904265974623366, 8.383427443423370046826559490756, 9.115993369649615658482087704142, 11.02569139384071029953096227765, 12.02648527071599864204168778327, 13.54697554992937124912407446891, 14.31229933959855414823111856959
