| L(s) = 1 | + 0.952·2-s + 2.51·3-s − 1.09·4-s + 5-s + 2.39·6-s − 7-s − 2.94·8-s + 3.33·9-s + 0.952·10-s − 4.99·11-s − 2.74·12-s + 5.08·13-s − 0.952·14-s + 2.51·15-s − 0.623·16-s − 0.789·17-s + 3.18·18-s − 6.69·19-s − 1.09·20-s − 2.51·21-s − 4.75·22-s + 5.29·23-s − 7.41·24-s + 25-s + 4.84·26-s + 0.855·27-s + 1.09·28-s + ⋯ |
| L(s) = 1 | + 0.673·2-s + 1.45·3-s − 0.546·4-s + 0.447·5-s + 0.979·6-s − 0.377·7-s − 1.04·8-s + 1.11·9-s + 0.301·10-s − 1.50·11-s − 0.793·12-s + 1.40·13-s − 0.254·14-s + 0.650·15-s − 0.155·16-s − 0.191·17-s + 0.750·18-s − 1.53·19-s − 0.244·20-s − 0.549·21-s − 1.01·22-s + 1.10·23-s − 1.51·24-s + 0.200·25-s + 0.949·26-s + 0.164·27-s + 0.206·28-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 | \( 1 - 0.952T + 2T^{2} \) |
| 3 | \( 1 - 2.51T + 3T^{2} \) |
| 11 | \( 1 + 4.99T + 11T^{2} \) |
| 13 | \( 1 - 5.08T + 13T^{2} \) |
| 17 | \( 1 + 0.789T + 17T^{2} \) |
| 19 | \( 1 + 6.69T + 19T^{2} \) |
| 23 | \( 1 - 5.29T + 23T^{2} \) |
| 29 | \( 1 + 4.72T + 29T^{2} \) |
| 31 | \( 1 + 3.65T + 31T^{2} \) |
| 37 | \( 1 + 0.793T + 37T^{2} \) |
| 41 | \( 1 + 6.05T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + 4.55T + 47T^{2} \) |
| 53 | \( 1 - 7.74T + 53T^{2} \) |
| 59 | \( 1 + 3.08T + 59T^{2} \) |
| 61 | \( 1 + 9.36T + 61T^{2} \) |
| 67 | \( 1 - 2.48T + 67T^{2} \) |
| 71 | \( 1 + 5.56T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 + 6.53T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 + 2.91T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−7.62664596136066558093254201818, −6.79034044674115960719462754072, −5.88498270752496103783272954034, −5.44264574289393787969796974419, −4.41236003774966180191181981362, −3.87159974389581385136136872132, −3.04493450933338391573748707690, −2.63264865526005610760370672919, −1.60995189833105488587105008400, 0,
1.60995189833105488587105008400, 2.63264865526005610760370672919, 3.04493450933338391573748707690, 3.87159974389581385136136872132, 4.41236003774966180191181981362, 5.44264574289393787969796974419, 5.88498270752496103783272954034, 6.79034044674115960719462754072, 7.62664596136066558093254201818
