L(s) = 1 | − 1.63·3-s + 2.87·5-s + 2.53·7-s − 0.317·9-s + 3.58·11-s + 5.49·13-s − 4.70·15-s + 6.64·17-s + 3.16·19-s − 4.15·21-s − 2.23·23-s + 3.25·25-s + 5.43·27-s + 5.29·29-s − 6.91·31-s − 5.86·33-s + 7.28·35-s + 6.97·37-s − 9.00·39-s + 8.56·41-s − 7.68·43-s − 0.912·45-s + 2.74·47-s − 0.577·49-s − 10.8·51-s − 6.46·53-s + 10.2·55-s + ⋯ |
L(s) = 1 | − 0.945·3-s + 1.28·5-s + 0.957·7-s − 0.105·9-s + 1.08·11-s + 1.52·13-s − 1.21·15-s + 1.61·17-s + 0.726·19-s − 0.905·21-s − 0.465·23-s + 0.651·25-s + 1.04·27-s + 0.983·29-s − 1.24·31-s − 1.02·33-s + 1.23·35-s + 1.14·37-s − 1.44·39-s + 1.33·41-s − 1.17·43-s − 0.136·45-s + 0.400·47-s − 0.0825·49-s − 1.52·51-s − 0.887·53-s + 1.38·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.518236444\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.518236444\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
good | 3 | \( 1 + 1.63T + 3T^{2} \) |
| 5 | \( 1 - 2.87T + 5T^{2} \) |
| 7 | \( 1 - 2.53T + 7T^{2} \) |
| 11 | \( 1 - 3.58T + 11T^{2} \) |
| 13 | \( 1 - 5.49T + 13T^{2} \) |
| 17 | \( 1 - 6.64T + 17T^{2} \) |
| 19 | \( 1 - 3.16T + 19T^{2} \) |
| 23 | \( 1 + 2.23T + 23T^{2} \) |
| 29 | \( 1 - 5.29T + 29T^{2} \) |
| 31 | \( 1 + 6.91T + 31T^{2} \) |
| 37 | \( 1 - 6.97T + 37T^{2} \) |
| 41 | \( 1 - 8.56T + 41T^{2} \) |
| 43 | \( 1 + 7.68T + 43T^{2} \) |
| 47 | \( 1 - 2.74T + 47T^{2} \) |
| 53 | \( 1 + 6.46T + 53T^{2} \) |
| 59 | \( 1 + 2.83T + 59T^{2} \) |
| 61 | \( 1 + 1.72T + 61T^{2} \) |
| 67 | \( 1 + 6.56T + 67T^{2} \) |
| 71 | \( 1 + 8.23T + 71T^{2} \) |
| 73 | \( 1 - 3.78T + 73T^{2} \) |
| 79 | \( 1 + 1.99T + 79T^{2} \) |
| 83 | \( 1 - 3.83T + 83T^{2} \) |
| 89 | \( 1 + 2.50T + 89T^{2} \) |
| 97 | \( 1 + 19.1T + 97T^{2} \) |
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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
−8.447812963050814930478629248293, −7.76791777090177741639848718985, −6.68629665119724066035406572397, −5.93229204559247803840337919227, −5.75427025141431172189907468826, −4.94065453329176072183166579304, −3.94731934012417733694528315789, −2.93082965242902212954169420030, −1.50182027629641387545992719374, −1.16666374012568492408831229437,
1.16666374012568492408831229437, 1.50182027629641387545992719374, 2.93082965242902212954169420030, 3.94731934012417733694528315789, 4.94065453329176072183166579304, 5.75427025141431172189907468826, 5.93229204559247803840337919227, 6.68629665119724066035406572397, 7.76791777090177741639848718985, 8.447812963050814930478629248293