| L(s) = 1 | + 1.98·2-s + 1.93·4-s − 2.87·5-s − 2.68·7-s − 0.124·8-s − 5.71·10-s − 0.368·11-s − 2.43·13-s − 5.32·14-s − 4.12·16-s + 3.34·17-s + 0.377·19-s − 5.57·20-s − 0.730·22-s + 0.915·23-s + 3.28·25-s − 4.83·26-s − 5.20·28-s + 2.06·29-s + 5.88·31-s − 7.92·32-s + 6.64·34-s + 7.72·35-s − 7.95·37-s + 0.749·38-s + 0.357·40-s + 1.24·41-s + ⋯ |
| L(s) = 1 | + 1.40·2-s + 0.968·4-s − 1.28·5-s − 1.01·7-s − 0.0438·8-s − 1.80·10-s − 0.110·11-s − 0.676·13-s − 1.42·14-s − 1.03·16-s + 0.811·17-s + 0.0866·19-s − 1.24·20-s − 0.155·22-s + 0.190·23-s + 0.657·25-s − 0.948·26-s − 0.982·28-s + 0.384·29-s + 1.05·31-s − 1.40·32-s + 1.13·34-s + 1.30·35-s − 1.30·37-s + 0.121·38-s + 0.0564·40-s + 0.193·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4923 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4923 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.074866589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.074866589\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 547 | \( 1 + T \) |
| good | 2 | \( 1 - 1.98T + 2T^{2} \) |
| 5 | \( 1 + 2.87T + 5T^{2} \) |
| 7 | \( 1 + 2.68T + 7T^{2} \) |
| 11 | \( 1 + 0.368T + 11T^{2} \) |
| 13 | \( 1 + 2.43T + 13T^{2} \) |
| 17 | \( 1 - 3.34T + 17T^{2} \) |
| 19 | \( 1 - 0.377T + 19T^{2} \) |
| 23 | \( 1 - 0.915T + 23T^{2} \) |
| 29 | \( 1 - 2.06T + 29T^{2} \) |
| 31 | \( 1 - 5.88T + 31T^{2} \) |
| 37 | \( 1 + 7.95T + 37T^{2} \) |
| 41 | \( 1 - 1.24T + 41T^{2} \) |
| 43 | \( 1 - 5.58T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 - 1.64T + 61T^{2} \) |
| 67 | \( 1 - 5.77T + 67T^{2} \) |
| 71 | \( 1 - 7.47T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 0.641T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 + 4.77T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
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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
−8.144378740972326683208337580901, −7.11785382963159677791613450889, −6.93861988646188814527195812772, −5.82676652048670388716732563352, −5.31147561796378483097417831338, −4.33091520831427119120616037680, −3.88313725679953150479889538072, −3.12781587612262431889057269245, −2.51093049391806525158446998810, −0.60892484578462612497927612763,
0.60892484578462612497927612763, 2.51093049391806525158446998810, 3.12781587612262431889057269245, 3.88313725679953150479889538072, 4.33091520831427119120616037680, 5.31147561796378483097417831338, 5.82676652048670388716732563352, 6.93861988646188814527195812772, 7.11785382963159677791613450889, 8.144378740972326683208337580901
