| L(s) = 1 | + 1.48·2-s + 0.193·4-s + 1.67·7-s − 2.67·8-s − 0.481·11-s + 1.28·13-s + 2.48·14-s − 4.35·16-s − 6.15·17-s − 19-s − 0.712·22-s + 2.54·23-s + 1.90·26-s + 0.324·28-s + 9.83·29-s − 9.50·31-s − 1.09·32-s − 9.11·34-s − 11.5·37-s − 1.48·38-s − 5.18·41-s − 2.06·43-s − 0.0933·44-s + 3.76·46-s − 5.76·47-s − 4.19·49-s + 0.249·52-s + ⋯ |
| L(s) = 1 | + 1.04·2-s + 0.0969·4-s + 0.633·7-s − 0.945·8-s − 0.145·11-s + 0.357·13-s + 0.663·14-s − 1.08·16-s − 1.49·17-s − 0.229·19-s − 0.151·22-s + 0.530·23-s + 0.373·26-s + 0.0613·28-s + 1.82·29-s − 1.70·31-s − 0.193·32-s − 1.56·34-s − 1.90·37-s − 0.240·38-s − 0.809·41-s − 0.314·43-s − 0.0140·44-s + 0.555·46-s − 0.841·47-s − 0.599·49-s + 0.0346·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 1.48T + 2T^{2} \) |
| 7 | \( 1 - 1.67T + 7T^{2} \) |
| 11 | \( 1 + 0.481T + 11T^{2} \) |
| 13 | \( 1 - 1.28T + 13T^{2} \) |
| 17 | \( 1 + 6.15T + 17T^{2} \) |
| 23 | \( 1 - 2.54T + 23T^{2} \) |
| 29 | \( 1 - 9.83T + 29T^{2} \) |
| 31 | \( 1 + 9.50T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 + 5.18T + 41T^{2} \) |
| 43 | \( 1 + 2.06T + 43T^{2} \) |
| 47 | \( 1 + 5.76T + 47T^{2} \) |
| 53 | \( 1 + 1.84T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 + 4.57T + 67T^{2} \) |
| 71 | \( 1 + 9.66T + 71T^{2} \) |
| 73 | \( 1 + 9.27T + 73T^{2} \) |
| 79 | \( 1 - 6.88T + 79T^{2} \) |
| 83 | \( 1 + 2.54T + 83T^{2} \) |
| 89 | \( 1 - 4.48T + 89T^{2} \) |
| 97 | \( 1 + 3.54T + 97T^{2} \) |
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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
−8.211557078316365253938479860397, −6.86741269512728489580878054710, −6.64467136067934004001563437154, −5.48748505664793921513872290235, −5.02952606563374110915778062879, −4.31626600088778584045283007768, −3.57323595315810526227348615013, −2.66261360717184704972546759033, −1.64745503230332527869258896499, 0,
1.64745503230332527869258896499, 2.66261360717184704972546759033, 3.57323595315810526227348615013, 4.31626600088778584045283007768, 5.02952606563374110915778062879, 5.48748505664793921513872290235, 6.64467136067934004001563437154, 6.86741269512728489580878054710, 8.211557078316365253938479860397
