| L(s) = 1 | + 0.554·2-s − 0.463·3-s − 1.69·4-s − 5-s − 0.256·6-s + 7-s − 2.04·8-s − 2.78·9-s − 0.554·10-s + 2.80·11-s + 0.784·12-s + 0.632·13-s + 0.554·14-s + 0.463·15-s + 2.25·16-s − 6.29·17-s − 1.54·18-s − 0.845·19-s + 1.69·20-s − 0.463·21-s + 1.55·22-s + 7.98·23-s + 0.948·24-s + 25-s + 0.350·26-s + 2.68·27-s − 1.69·28-s + ⋯ |
| L(s) = 1 | + 0.392·2-s − 0.267·3-s − 0.846·4-s − 0.447·5-s − 0.104·6-s + 0.377·7-s − 0.723·8-s − 0.928·9-s − 0.175·10-s + 0.844·11-s + 0.226·12-s + 0.175·13-s + 0.148·14-s + 0.119·15-s + 0.562·16-s − 1.52·17-s − 0.363·18-s − 0.193·19-s + 0.378·20-s − 0.101·21-s + 0.330·22-s + 1.66·23-s + 0.193·24-s + 0.200·25-s + 0.0687·26-s + 0.516·27-s − 0.319·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 - 0.554T + 2T^{2} \) |
| 3 | \( 1 + 0.463T + 3T^{2} \) |
| 11 | \( 1 - 2.80T + 11T^{2} \) |
| 13 | \( 1 - 0.632T + 13T^{2} \) |
| 17 | \( 1 + 6.29T + 17T^{2} \) |
| 19 | \( 1 + 0.845T + 19T^{2} \) |
| 23 | \( 1 - 7.98T + 23T^{2} \) |
| 29 | \( 1 + 0.164T + 29T^{2} \) |
| 31 | \( 1 - 1.38T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 + 3.30T + 41T^{2} \) |
| 43 | \( 1 - 3.01T + 43T^{2} \) |
| 47 | \( 1 - 3.86T + 47T^{2} \) |
| 53 | \( 1 - 4.44T + 53T^{2} \) |
| 59 | \( 1 - 4.31T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 2.57T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 2.28T + 73T^{2} \) |
| 79 | \( 1 - 2.66T + 79T^{2} \) |
| 83 | \( 1 + 0.742T + 83T^{2} \) |
| 89 | \( 1 - 5.80T + 89T^{2} \) |
| 97 | \( 1 - 6.81T + 97T^{2} \) |
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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
−7.38831467449086163983284030728, −6.70629634228505985112550407470, −6.04716061537370688138881026651, −5.17317197278562737031919152411, −4.76812528269209833878796868167, −3.95683870110195086783955339816, −3.33873869322464106517450930560, −2.37137775851616020411920132295, −1.04764844981442911523898783125, 0,
1.04764844981442911523898783125, 2.37137775851616020411920132295, 3.33873869322464106517450930560, 3.95683870110195086783955339816, 4.76812528269209833878796868167, 5.17317197278562737031919152411, 6.04716061537370688138881026651, 6.70629634228505985112550407470, 7.38831467449086163983284030728
