| L(s) = 1 | + 0.184·2-s − 3-s − 1.96·4-s − 0.184·6-s + 1.90·7-s − 0.733·8-s + 9-s − 3.94·11-s + 1.96·12-s + 4.53·13-s + 0.353·14-s + 3.79·16-s − 4.68·17-s + 0.184·18-s + 19-s − 1.90·21-s − 0.728·22-s − 6.18·23-s + 0.733·24-s + 0.838·26-s − 27-s − 3.75·28-s + 5.98·29-s + 7.31·31-s + 2.16·32-s + 3.94·33-s − 0.866·34-s + ⋯ |
| L(s) = 1 | + 0.130·2-s − 0.577·3-s − 0.982·4-s − 0.0754·6-s + 0.721·7-s − 0.259·8-s + 0.333·9-s − 1.18·11-s + 0.567·12-s + 1.25·13-s + 0.0943·14-s + 0.949·16-s − 1.13·17-s + 0.0435·18-s + 0.229·19-s − 0.416·21-s − 0.155·22-s − 1.28·23-s + 0.149·24-s + 0.164·26-s − 0.192·27-s − 0.709·28-s + 1.11·29-s + 1.31·31-s + 0.383·32-s + 0.685·33-s − 0.148·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 0.184T + 2T^{2} \) |
| 7 | \( 1 - 1.90T + 7T^{2} \) |
| 11 | \( 1 + 3.94T + 11T^{2} \) |
| 13 | \( 1 - 4.53T + 13T^{2} \) |
| 17 | \( 1 + 4.68T + 17T^{2} \) |
| 23 | \( 1 + 6.18T + 23T^{2} \) |
| 29 | \( 1 - 5.98T + 29T^{2} \) |
| 31 | \( 1 - 7.31T + 31T^{2} \) |
| 37 | \( 1 + 8.07T + 37T^{2} \) |
| 41 | \( 1 - 0.0567T + 41T^{2} \) |
| 43 | \( 1 - 0.822T + 43T^{2} \) |
| 47 | \( 1 + 7.50T + 47T^{2} \) |
| 53 | \( 1 - 1.28T + 53T^{2} \) |
| 59 | \( 1 + 8.28T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 - 4.12T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 - 7.17T + 89T^{2} \) |
| 97 | \( 1 - 2.70T + 97T^{2} \) |
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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
−8.992707608264709699909328726113, −8.343710579607143504058382277608, −7.76305316692633682426891102573, −6.44472032117261233550590023470, −5.74839603913227173594103882881, −4.79378241502298448255287399396, −4.34172587121247437822643043702, −3.07424410882179948394472030029, −1.52409567709379873437860932791, 0,
1.52409567709379873437860932791, 3.07424410882179948394472030029, 4.34172587121247437822643043702, 4.79378241502298448255287399396, 5.74839603913227173594103882881, 6.44472032117261233550590023470, 7.76305316692633682426891102573, 8.343710579607143504058382277608, 8.992707608264709699909328726113
