L(s) = 1 | − 1.83·2-s + 1.40·3-s + 1.38·4-s + 5-s − 2.57·6-s + 7-s + 1.13·8-s − 1.03·9-s − 1.83·10-s − 2.97·11-s + 1.93·12-s + 3.06·13-s − 1.83·14-s + 1.40·15-s − 4.85·16-s + 1.75·17-s + 1.91·18-s + 4.20·19-s + 1.38·20-s + 1.40·21-s + 5.46·22-s − 2.41·23-s + 1.59·24-s + 25-s − 5.64·26-s − 5.65·27-s + 1.38·28-s + ⋯ |
L(s) = 1 | − 1.30·2-s + 0.808·3-s + 0.690·4-s + 0.447·5-s − 1.05·6-s + 0.377·7-s + 0.402·8-s − 0.346·9-s − 0.581·10-s − 0.896·11-s + 0.557·12-s + 0.851·13-s − 0.491·14-s + 0.361·15-s − 1.21·16-s + 0.425·17-s + 0.450·18-s + 0.965·19-s + 0.308·20-s + 0.305·21-s + 1.16·22-s − 0.502·23-s + 0.325·24-s + 0.200·25-s − 1.10·26-s − 1.08·27-s + 0.260·28-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 + T \) |
good | 2 | \( 1 + 1.83T + 2T^{2} \) |
| 3 | \( 1 - 1.40T + 3T^{2} \) |
| 11 | \( 1 + 2.97T + 11T^{2} \) |
| 13 | \( 1 - 3.06T + 13T^{2} \) |
| 17 | \( 1 - 1.75T + 17T^{2} \) |
| 19 | \( 1 - 4.20T + 19T^{2} \) |
| 23 | \( 1 + 2.41T + 23T^{2} \) |
| 29 | \( 1 + 6.62T + 29T^{2} \) |
| 31 | \( 1 + 0.754T + 31T^{2} \) |
| 37 | \( 1 + 1.80T + 37T^{2} \) |
| 41 | \( 1 - 8.96T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 2.64T + 47T^{2} \) |
| 53 | \( 1 + 4.13T + 53T^{2} \) |
| 59 | \( 1 + 9.55T + 59T^{2} \) |
| 61 | \( 1 + 4.65T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 - 9.25T + 71T^{2} \) |
| 73 | \( 1 + 8.65T + 73T^{2} \) |
| 79 | \( 1 + 9.11T + 79T^{2} \) |
| 83 | \( 1 - 1.29T + 83T^{2} \) |
| 89 | \( 1 - 0.189T + 89T^{2} \) |
| 97 | \( 1 - 1.37T + 97T^{2} \) |
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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
−7.70521541277171271382455276195, −7.32783681477226817685540672092, −6.13745768537817113111412194168, −5.54668642736608096975056113551, −4.70219374919032804940961561050, −3.64058077699730845570305744395, −2.88143899899452369998352565677, −1.99149665188553652528262572569, −1.30612632100285933772405426825, 0,
1.30612632100285933772405426825, 1.99149665188553652528262572569, 2.88143899899452369998352565677, 3.64058077699730845570305744395, 4.70219374919032804940961561050, 5.54668642736608096975056113551, 6.13745768537817113111412194168, 7.32783681477226817685540672092, 7.70521541277171271382455276195