| L(s) = 1 | + 1.46·2-s − 2.95·3-s + 0.150·4-s + 3.07·5-s − 4.33·6-s − 3.08·7-s − 2.71·8-s + 5.72·9-s + 4.50·10-s + 2.74·11-s − 0.445·12-s − 2.20·13-s − 4.52·14-s − 9.08·15-s − 4.27·16-s − 7.63·17-s + 8.40·18-s − 0.0323·19-s + 0.463·20-s + 9.12·21-s + 4.03·22-s − 6.39·23-s + 8.01·24-s + 4.45·25-s − 3.23·26-s − 8.06·27-s − 0.465·28-s + ⋯ |
| L(s) = 1 | + 1.03·2-s − 1.70·3-s + 0.0753·4-s + 1.37·5-s − 1.76·6-s − 1.16·7-s − 0.958·8-s + 1.90·9-s + 1.42·10-s + 0.829·11-s − 0.128·12-s − 0.611·13-s − 1.21·14-s − 2.34·15-s − 1.06·16-s − 1.85·17-s + 1.98·18-s − 0.00742·19-s + 0.103·20-s + 1.99·21-s + 0.859·22-s − 1.33·23-s + 1.63·24-s + 0.890·25-s − 0.633·26-s − 1.55·27-s − 0.0879·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9592531138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9592531138\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 97 | \( 1 \) |
| good | 2 | \( 1 - 1.46T + 2T^{2} \) |
| 3 | \( 1 + 2.95T + 3T^{2} \) |
| 5 | \( 1 - 3.07T + 5T^{2} \) |
| 7 | \( 1 + 3.08T + 7T^{2} \) |
| 11 | \( 1 - 2.74T + 11T^{2} \) |
| 13 | \( 1 + 2.20T + 13T^{2} \) |
| 17 | \( 1 + 7.63T + 17T^{2} \) |
| 19 | \( 1 + 0.0323T + 19T^{2} \) |
| 23 | \( 1 + 6.39T + 23T^{2} \) |
| 29 | \( 1 - 0.734T + 29T^{2} \) |
| 31 | \( 1 - 4.24T + 31T^{2} \) |
| 37 | \( 1 - 0.467T + 37T^{2} \) |
| 41 | \( 1 + 8.88T + 41T^{2} \) |
| 43 | \( 1 - 5.73T + 43T^{2} \) |
| 47 | \( 1 - 0.108T + 47T^{2} \) |
| 53 | \( 1 + 8.55T + 53T^{2} \) |
| 59 | \( 1 - 6.44T + 59T^{2} \) |
| 61 | \( 1 - 1.22T + 61T^{2} \) |
| 67 | \( 1 + 6.85T + 67T^{2} \) |
| 71 | \( 1 - 5.16T + 71T^{2} \) |
| 73 | \( 1 - 8.09T + 73T^{2} \) |
| 79 | \( 1 + 0.275T + 79T^{2} \) |
| 83 | \( 1 + 6.28T + 83T^{2} \) |
| 89 | \( 1 - 6.33T + 89T^{2} \) |
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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
−7.01788324133732115393523079472, −6.50789118660236893445542836026, −6.23591035783484202826993637464, −5.76892866258319234931660916901, −4.99414311666791976551087589471, −4.47604877303592472806214886429, −3.76540945213815230362946852265, −2.64199533825870647609414995180, −1.80871710601550487305351179573, −0.42043062885842925559709266251,
0.42043062885842925559709266251, 1.80871710601550487305351179573, 2.64199533825870647609414995180, 3.76540945213815230362946852265, 4.47604877303592472806214886429, 4.99414311666791976551087589471, 5.76892866258319234931660916901, 6.23591035783484202826993637464, 6.50789118660236893445542836026, 7.01788324133732115393523079472
