| L(s) = 1 | + 2-s − 1.42·3-s + 4-s − 4.18·5-s − 1.42·6-s − 7-s + 8-s − 0.982·9-s − 4.18·10-s − 2.90·11-s − 1.42·12-s + 0.795·13-s − 14-s + 5.95·15-s + 16-s − 1.40·17-s − 0.982·18-s + 5.53·19-s − 4.18·20-s + 1.42·21-s − 2.90·22-s + 7.61·23-s − 1.42·24-s + 12.5·25-s + 0.795·26-s + 5.65·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.820·3-s + 0.5·4-s − 1.87·5-s − 0.579·6-s − 0.377·7-s + 0.353·8-s − 0.327·9-s − 1.32·10-s − 0.875·11-s − 0.410·12-s + 0.220·13-s − 0.267·14-s + 1.53·15-s + 0.250·16-s − 0.340·17-s − 0.231·18-s + 1.27·19-s − 0.936·20-s + 0.309·21-s − 0.619·22-s + 1.58·23-s − 0.289·24-s + 2.51·25-s + 0.156·26-s + 1.08·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 + 1.42T + 3T^{2} \) |
| 5 | \( 1 + 4.18T + 5T^{2} \) |
| 11 | \( 1 + 2.90T + 11T^{2} \) |
| 13 | \( 1 - 0.795T + 13T^{2} \) |
| 17 | \( 1 + 1.40T + 17T^{2} \) |
| 19 | \( 1 - 5.53T + 19T^{2} \) |
| 23 | \( 1 - 7.61T + 23T^{2} \) |
| 29 | \( 1 - 1.05T + 29T^{2} \) |
| 31 | \( 1 - 3.96T + 31T^{2} \) |
| 37 | \( 1 + 4.58T + 37T^{2} \) |
| 41 | \( 1 + 5.80T + 41T^{2} \) |
| 43 | \( 1 - 2.10T + 43T^{2} \) |
| 47 | \( 1 + 9.67T + 47T^{2} \) |
| 53 | \( 1 - 8.61T + 53T^{2} \) |
| 59 | \( 1 + 8.46T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 + 5.27T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 - 6.96T + 73T^{2} \) |
| 79 | \( 1 - 7.77T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 + 0.471T + 89T^{2} \) |
| 97 | \( 1 + 5.30T + 97T^{2} \) |
| 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
−7.48066168620527857220385567287, −7.01704828433476458085633838777, −6.33025831423005016169336418701, −5.21665834986727041905995221148, −5.02839896127041177169415177594, −4.11332271473997626555300054205, −3.20333479565226340823576527997, −2.87753439145892419356444094614, −1.00247040969992397494363531129, 0,
1.00247040969992397494363531129, 2.87753439145892419356444094614, 3.20333479565226340823576527997, 4.11332271473997626555300054205, 5.02839896127041177169415177594, 5.21665834986727041905995221148, 6.33025831423005016169336418701, 7.01704828433476458085633838777, 7.48066168620527857220385567287
