| L(s) = 1 | + 1.47·3-s − 1.11·7-s − 0.830·9-s − 2.22·11-s − 1.47·13-s + 4.06·17-s + 5.51·19-s − 1.64·21-s − 1.24·23-s − 5.64·27-s − 29-s + 1.83·31-s − 3.28·33-s − 1.05·37-s − 2.16·39-s − 4.22·41-s − 7.83·43-s − 2.71·47-s − 5.75·49-s + 5.98·51-s − 9.34·53-s + 8.12·57-s + 0.904·59-s + 13.2·61-s + 0.926·63-s − 1.43·67-s − 1.83·69-s + ⋯ |
| L(s) = 1 | + 0.850·3-s − 0.421·7-s − 0.276·9-s − 0.672·11-s − 0.408·13-s + 0.984·17-s + 1.26·19-s − 0.358·21-s − 0.259·23-s − 1.08·27-s − 0.185·29-s + 0.328·31-s − 0.571·33-s − 0.173·37-s − 0.347·39-s − 0.660·41-s − 1.19·43-s − 0.396·47-s − 0.822·49-s + 0.837·51-s − 1.28·53-s + 1.07·57-s + 0.117·59-s + 1.69·61-s + 0.116·63-s − 0.174·67-s − 0.220·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 1.47T + 3T^{2} \) |
| 7 | \( 1 + 1.11T + 7T^{2} \) |
| 11 | \( 1 + 2.22T + 11T^{2} \) |
| 13 | \( 1 + 1.47T + 13T^{2} \) |
| 17 | \( 1 - 4.06T + 17T^{2} \) |
| 19 | \( 1 - 5.51T + 19T^{2} \) |
| 23 | \( 1 + 1.24T + 23T^{2} \) |
| 31 | \( 1 - 1.83T + 31T^{2} \) |
| 37 | \( 1 + 1.05T + 37T^{2} \) |
| 41 | \( 1 + 4.22T + 41T^{2} \) |
| 43 | \( 1 + 7.83T + 43T^{2} \) |
| 47 | \( 1 + 2.71T + 47T^{2} \) |
| 53 | \( 1 + 9.34T + 53T^{2} \) |
| 59 | \( 1 - 0.904T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 + 1.43T + 67T^{2} \) |
| 71 | \( 1 - 6.56T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 - 8.98T + 79T^{2} \) |
| 83 | \( 1 + 6.94T + 83T^{2} \) |
| 89 | \( 1 + 3.05T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−7.941382828154460178518240569631, −7.21377999081794741909835002105, −6.39360646014050570864186769436, −5.45079686099396517277262396832, −5.02281687951258769320880715368, −3.75615631073097299164817546329, −3.17129029307589156238009323320, −2.57474627009798986336706543904, −1.46630934825438658559288289812, 0,
1.46630934825438658559288289812, 2.57474627009798986336706543904, 3.17129029307589156238009323320, 3.75615631073097299164817546329, 5.02281687951258769320880715368, 5.45079686099396517277262396832, 6.39360646014050570864186769436, 7.21377999081794741909835002105, 7.941382828154460178518240569631
