| L(s) = 1 | + 2.30·2-s − 1.69·3-s + 3.30·4-s − 5-s − 3.90·6-s + 7-s + 3.00·8-s − 0.122·9-s − 2.30·10-s + 2.44·11-s − 5.60·12-s − 3.07·13-s + 2.30·14-s + 1.69·15-s + 0.316·16-s + 4.65·17-s − 0.281·18-s − 1.53·19-s − 3.30·20-s − 1.69·21-s + 5.62·22-s − 6.77·23-s − 5.10·24-s + 25-s − 7.08·26-s + 5.29·27-s + 3.30·28-s + ⋯ |
| L(s) = 1 | + 1.62·2-s − 0.979·3-s + 1.65·4-s − 0.447·5-s − 1.59·6-s + 0.377·7-s + 1.06·8-s − 0.0407·9-s − 0.728·10-s + 0.736·11-s − 1.61·12-s − 0.853·13-s + 0.615·14-s + 0.438·15-s + 0.0791·16-s + 1.12·17-s − 0.0662·18-s − 0.352·19-s − 0.739·20-s − 0.370·21-s + 1.19·22-s − 1.41·23-s − 1.04·24-s + 0.200·25-s − 1.38·26-s + 1.01·27-s + 0.624·28-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 | \( 1 - 2.30T + 2T^{2} \) |
| 3 | \( 1 + 1.69T + 3T^{2} \) |
| 11 | \( 1 - 2.44T + 11T^{2} \) |
| 13 | \( 1 + 3.07T + 13T^{2} \) |
| 17 | \( 1 - 4.65T + 17T^{2} \) |
| 19 | \( 1 + 1.53T + 19T^{2} \) |
| 23 | \( 1 + 6.77T + 23T^{2} \) |
| 29 | \( 1 - 5.36T + 29T^{2} \) |
| 31 | \( 1 - 6.79T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 + 3.44T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 - 4.66T + 47T^{2} \) |
| 53 | \( 1 + 3.19T + 53T^{2} \) |
| 59 | \( 1 - 7.63T + 59T^{2} \) |
| 61 | \( 1 + 0.573T + 61T^{2} \) |
| 67 | \( 1 - 9.21T + 67T^{2} \) |
| 71 | \( 1 + 4.36T + 71T^{2} \) |
| 73 | \( 1 - 4.83T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 - 6.31T + 97T^{2} \) |
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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
−6.90910406299297798474974189783, −6.72175766819591055071814229534, −5.82821740115570910128319486271, −5.34197798637319947639922889537, −4.73259231618032806087501097245, −4.12214701882809818708994205391, −3.35808134533829760345385884269, −2.54129765187407538564866533255, −1.41746597732031730583163335501, 0,
1.41746597732031730583163335501, 2.54129765187407538564866533255, 3.35808134533829760345385884269, 4.12214701882809818708994205391, 4.73259231618032806087501097245, 5.34197798637319947639922889537, 5.82821740115570910128319486271, 6.72175766819591055071814229534, 6.90910406299297798474974189783
