| L(s) = 1 | − 2.35·2-s + 1.42·3-s + 3.53·4-s − 5-s − 3.34·6-s − 7-s − 3.61·8-s − 0.978·9-s + 2.35·10-s − 5.71·11-s + 5.02·12-s − 2.54·13-s + 2.35·14-s − 1.42·15-s + 1.43·16-s − 3.84·17-s + 2.30·18-s + 4.40·19-s − 3.53·20-s − 1.42·21-s + 13.4·22-s − 5.65·23-s − 5.14·24-s + 25-s + 5.98·26-s − 5.65·27-s − 3.53·28-s + ⋯ |
| L(s) = 1 | − 1.66·2-s + 0.820·3-s + 1.76·4-s − 0.447·5-s − 1.36·6-s − 0.377·7-s − 1.27·8-s − 0.326·9-s + 0.744·10-s − 1.72·11-s + 1.45·12-s − 0.705·13-s + 0.628·14-s − 0.367·15-s + 0.358·16-s − 0.933·17-s + 0.542·18-s + 1.01·19-s − 0.790·20-s − 0.310·21-s + 2.86·22-s − 1.17·23-s − 1.04·24-s + 0.200·25-s + 1.17·26-s − 1.08·27-s − 0.668·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1185456727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1185456727\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 + 2.35T + 2T^{2} \) |
| 3 | \( 1 - 1.42T + 3T^{2} \) |
| 11 | \( 1 + 5.71T + 11T^{2} \) |
| 13 | \( 1 + 2.54T + 13T^{2} \) |
| 17 | \( 1 + 3.84T + 17T^{2} \) |
| 19 | \( 1 - 4.40T + 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 + 6.59T + 29T^{2} \) |
| 31 | \( 1 + 1.55T + 31T^{2} \) |
| 37 | \( 1 + 1.11T + 37T^{2} \) |
| 41 | \( 1 + 6.51T + 41T^{2} \) |
| 43 | \( 1 - 8.12T + 43T^{2} \) |
| 47 | \( 1 + 7.20T + 47T^{2} \) |
| 53 | \( 1 - 1.36T + 53T^{2} \) |
| 59 | \( 1 + 3.51T + 59T^{2} \) |
| 61 | \( 1 - 9.88T + 61T^{2} \) |
| 67 | \( 1 - 9.52T + 67T^{2} \) |
| 71 | \( 1 + 6.37T + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 + 15.9T + 79T^{2} \) |
| 83 | \( 1 + 5.46T + 83T^{2} \) |
| 89 | \( 1 + 1.58T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.991746893437294385006608843970, −7.44438477251168789829272090295, −7.01826113752619168015112648395, −5.89327735276497983486251779824, −5.18795323384909543241297936779, −4.07544999468850695582759678417, −3.04093094583290215867033889531, −2.51798321459127111927564930023, −1.79365738627434685681233876763, −0.19498882472648747044158776376,
0.19498882472648747044158776376, 1.79365738627434685681233876763, 2.51798321459127111927564930023, 3.04093094583290215867033889531, 4.07544999468850695582759678417, 5.18795323384909543241297936779, 5.89327735276497983486251779824, 7.01826113752619168015112648395, 7.44438477251168789829272090295, 7.991746893437294385006608843970
