| L(s) = 1 | + 2·2-s + 3-s + 4·4-s + 2·6-s − 7·7-s + 8·8-s − 26·9-s − 65·11-s + 4·12-s − 13·13-s − 14·14-s + 16·16-s + 73·17-s − 52·18-s − 142·19-s − 7·21-s − 130·22-s − 130·23-s + 8·24-s − 26·26-s − 53·27-s − 28·28-s + 111·29-s + 256·31-s + 32·32-s − 65·33-s + 146·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.192·3-s + 1/2·4-s + 0.136·6-s − 0.377·7-s + 0.353·8-s − 0.962·9-s − 1.78·11-s + 0.0962·12-s − 0.277·13-s − 0.267·14-s + 1/4·16-s + 1.04·17-s − 0.680·18-s − 1.71·19-s − 0.0727·21-s − 1.25·22-s − 1.17·23-s + 0.0680·24-s − 0.196·26-s − 0.377·27-s − 0.188·28-s + 0.710·29-s + 1.48·31-s + 0.176·32-s − 0.342·33-s + 0.736·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 3 | \( 1 - T + p^{3} T^{2} \) |
| 11 | \( 1 + 65 T + p^{3} T^{2} \) |
| 13 | \( 1 + p T + p^{3} T^{2} \) |
| 17 | \( 1 - 73 T + p^{3} T^{2} \) |
| 19 | \( 1 + 142 T + p^{3} T^{2} \) |
| 23 | \( 1 + 130 T + p^{3} T^{2} \) |
| 29 | \( 1 - 111 T + p^{3} T^{2} \) |
| 31 | \( 1 - 256 T + p^{3} T^{2} \) |
| 37 | \( 1 - 266 T + p^{3} T^{2} \) |
| 41 | \( 1 + 424 T + p^{3} T^{2} \) |
| 43 | \( 1 + 534 T + p^{3} T^{2} \) |
| 47 | \( 1 - 269 T + p^{3} T^{2} \) |
| 53 | \( 1 - 132 T + p^{3} T^{2} \) |
| 59 | \( 1 + 224 T + p^{3} T^{2} \) |
| 61 | \( 1 + 572 T + p^{3} T^{2} \) |
| 67 | \( 1 - 108 T + p^{3} T^{2} \) |
| 71 | \( 1 - 560 T + p^{3} T^{2} \) |
| 73 | \( 1 + 586 T + p^{3} T^{2} \) |
| 79 | \( 1 - 57 T + p^{3} T^{2} \) |
| 83 | \( 1 + 252 T + p^{3} T^{2} \) |
| 89 | \( 1 + 184 T + p^{3} T^{2} \) |
| 97 | \( 1 - 605 T + p^{3} T^{2} \) |
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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
−10.51202538660914423337880245902, −10.02303496236146344813180028657, −8.397555422907168281580591682920, −7.904120902346070149091163566062, −6.47945949906103912958512390862, −5.63212165100733295772054731878, −4.61451052827223941136579288238, −3.17607081277176285311919478287, −2.33910046513794241948102446195, 0,
2.33910046513794241948102446195, 3.17607081277176285311919478287, 4.61451052827223941136579288238, 5.63212165100733295772054731878, 6.47945949906103912958512390862, 7.904120902346070149091163566062, 8.397555422907168281580591682920, 10.02303496236146344813180028657, 10.51202538660914423337880245902
