| L(s) = 1 | + 1.94·2-s − 3·3-s − 4.23·4-s + 13.4·5-s − 5.82·6-s − 31.5·7-s − 23.7·8-s + 9·9-s + 26.0·10-s + 11·11-s + 12.7·12-s − 8.59·13-s − 61.2·14-s − 40.2·15-s − 12.1·16-s + 59.9·17-s + 17.4·18-s + 67.3·19-s − 56.8·20-s + 94.6·21-s + 21.3·22-s − 47.9·23-s + 71.2·24-s + 55.2·25-s − 16.6·26-s − 27·27-s + 133.·28-s + ⋯ |
| L(s) = 1 | + 0.685·2-s − 0.577·3-s − 0.529·4-s + 1.20·5-s − 0.396·6-s − 1.70·7-s − 1.04·8-s + 0.333·9-s + 0.823·10-s + 0.301·11-s + 0.305·12-s − 0.183·13-s − 1.16·14-s − 0.693·15-s − 0.190·16-s + 0.854·17-s + 0.228·18-s + 0.813·19-s − 0.635·20-s + 0.983·21-s + 0.206·22-s − 0.434·23-s + 0.605·24-s + 0.441·25-s − 0.125·26-s − 0.192·27-s + 0.901·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 - 1.94T + 8T^{2} \) |
| 5 | \( 1 - 13.4T + 125T^{2} \) |
| 7 | \( 1 + 31.5T + 343T^{2} \) |
| 13 | \( 1 + 8.59T + 2.19e3T^{2} \) |
| 17 | \( 1 - 59.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 67.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 47.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 284.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 166.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 0.0469T + 5.06e4T^{2} \) |
| 41 | \( 1 - 68.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 196.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 394.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 130.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 433.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 212.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 443.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.10e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 552.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.18e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 397.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 885.T + 9.12e5T^{2} \) |
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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
−8.693116767192343117687210147540, −7.29395389524595899198723991796, −6.46534999525665616030093192602, −5.84884761909217016499889341874, −5.45572586756275437156619054598, −4.36841574387208040539613734617, −3.40525174868643304241888864869, −2.68534564711454859324612067552, −1.14235834341909326724545122225, 0,
1.14235834341909326724545122225, 2.68534564711454859324612067552, 3.40525174868643304241888864869, 4.36841574387208040539613734617, 5.45572586756275437156619054598, 5.84884761909217016499889341874, 6.46534999525665616030093192602, 7.29395389524595899198723991796, 8.693116767192343117687210147540
