| L(s) = 1 | + 2.69·2-s − 1.59·3-s + 5.28·4-s + 5-s − 4.29·6-s + 0.342·7-s + 8.86·8-s − 0.468·9-s + 2.69·10-s + 11-s − 8.40·12-s + 4.11·13-s + 0.924·14-s − 1.59·15-s + 13.3·16-s − 2.11·17-s − 1.26·18-s + 19-s + 5.28·20-s − 0.545·21-s + 2.69·22-s − 3.85·23-s − 14.0·24-s + 25-s + 11.0·26-s + 5.51·27-s + 1.81·28-s + ⋯ |
| L(s) = 1 | + 1.90·2-s − 0.918·3-s + 2.64·4-s + 0.447·5-s − 1.75·6-s + 0.129·7-s + 3.13·8-s − 0.156·9-s + 0.853·10-s + 0.301·11-s − 2.42·12-s + 1.14·13-s + 0.247·14-s − 0.410·15-s + 3.33·16-s − 0.512·17-s − 0.297·18-s + 0.229·19-s + 1.18·20-s − 0.119·21-s + 0.575·22-s − 0.804·23-s − 2.87·24-s + 0.200·25-s + 2.17·26-s + 1.06·27-s + 0.342·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.360044532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.360044532\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.69T + 2T^{2} \) |
| 3 | \( 1 + 1.59T + 3T^{2} \) |
| 7 | \( 1 - 0.342T + 7T^{2} \) |
| 13 | \( 1 - 4.11T + 13T^{2} \) |
| 17 | \( 1 + 2.11T + 17T^{2} \) |
| 23 | \( 1 + 3.85T + 23T^{2} \) |
| 29 | \( 1 + 2.82T + 29T^{2} \) |
| 31 | \( 1 - 0.00585T + 31T^{2} \) |
| 37 | \( 1 + 1.43T + 37T^{2} \) |
| 41 | \( 1 - 5.67T + 41T^{2} \) |
| 43 | \( 1 - 1.32T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 - 2.23T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 - 4.61T + 71T^{2} \) |
| 73 | \( 1 + 7.55T + 73T^{2} \) |
| 79 | \( 1 + 9.55T + 79T^{2} \) |
| 83 | \( 1 - 8.96T + 83T^{2} \) |
| 89 | \( 1 - 9.38T + 89T^{2} \) |
| 97 | \( 1 - 3.61T + 97T^{2} \) |
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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
−10.59542645478145340122461193778, −9.183177714853750453842028432648, −7.903885845933649149178261841465, −6.80999858094840278662488810298, −6.05918097642049535428523745738, −5.74104382420089527325508954801, −4.75394949883438994253653776501, −3.93700056396875952031645388980, −2.84649698042581928235324574985, −1.56893248779310571544452766829,
1.56893248779310571544452766829, 2.84649698042581928235324574985, 3.93700056396875952031645388980, 4.75394949883438994253653776501, 5.74104382420089527325508954801, 6.05918097642049535428523745738, 6.80999858094840278662488810298, 7.903885845933649149178261841465, 9.183177714853750453842028432648, 10.59542645478145340122461193778
