L(s) = 1 | − 2-s + 1.34·3-s + 4-s + 5-s − 1.34·6-s + 0.948·7-s − 8-s − 1.19·9-s − 10-s + 2.71·11-s + 1.34·12-s + 2.35·13-s − 0.948·14-s + 1.34·15-s + 16-s + 0.966·17-s + 1.19·18-s + 7.96·19-s + 20-s + 1.27·21-s − 2.71·22-s − 1.34·24-s + 25-s − 2.35·26-s − 5.63·27-s + 0.948·28-s + 10.3·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.776·3-s + 0.5·4-s + 0.447·5-s − 0.548·6-s + 0.358·7-s − 0.353·8-s − 0.397·9-s − 0.316·10-s + 0.817·11-s + 0.388·12-s + 0.651·13-s − 0.253·14-s + 0.347·15-s + 0.250·16-s + 0.234·17-s + 0.281·18-s + 1.82·19-s + 0.223·20-s + 0.278·21-s − 0.578·22-s − 0.274·24-s + 0.200·25-s − 0.460·26-s − 1.08·27-s + 0.179·28-s + 1.92·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.489508697\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.489508697\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 - 1.34T + 3T^{2} \) |
| 7 | \( 1 - 0.948T + 7T^{2} \) |
| 11 | \( 1 - 2.71T + 11T^{2} \) |
| 13 | \( 1 - 2.35T + 13T^{2} \) |
| 17 | \( 1 - 0.966T + 17T^{2} \) |
| 19 | \( 1 - 7.96T + 19T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 - 2.62T + 31T^{2} \) |
| 37 | \( 1 - 3.71T + 37T^{2} \) |
| 41 | \( 1 + 1.84T + 41T^{2} \) |
| 43 | \( 1 + 5.08T + 43T^{2} \) |
| 47 | \( 1 + 1.77T + 47T^{2} \) |
| 53 | \( 1 + 0.594T + 53T^{2} \) |
| 59 | \( 1 + 1.24T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 - 1.91T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 + 3.25T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 + 1.09T + 89T^{2} \) |
| 97 | \( 1 + 0.767T + 97T^{2} \) |
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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
−8.306934929807837211653262211129, −7.74181526036591217347925428474, −6.85332431612210667467599194907, −6.18683414999740425704438330472, −5.42287840032292066949063370339, −4.47170127265948458334461037645, −3.29289129496953358925057485927, −2.90569008306968395026247022939, −1.72331712087508928882623001456, −0.990267177017627222786460777582,
0.990267177017627222786460777582, 1.72331712087508928882623001456, 2.90569008306968395026247022939, 3.29289129496953358925057485927, 4.47170127265948458334461037645, 5.42287840032292066949063370339, 6.18683414999740425704438330472, 6.85332431612210667467599194907, 7.74181526036591217347925428474, 8.306934929807837211653262211129