| L(s) = 1 | + 1.79·5-s + 7-s − 0.736·11-s − 5.06·13-s + 3.33·17-s + 19-s − 7.06·23-s − 1.79·25-s − 5.27·29-s − 7.80·31-s + 1.79·35-s + 8.33·37-s + 5.58·41-s + 7.84·47-s + 49-s + 1.27·53-s − 1.31·55-s − 6.90·59-s + 6·61-s − 9.08·65-s + 14.7·67-s + 12.0·71-s + 9.87·73-s − 0.736·77-s − 0.805·79-s + 9.38·83-s + 5.97·85-s + ⋯ |
| L(s) = 1 | + 0.801·5-s + 0.377·7-s − 0.222·11-s − 1.40·13-s + 0.808·17-s + 0.229·19-s − 1.47·23-s − 0.358·25-s − 0.979·29-s − 1.40·31-s + 0.302·35-s + 1.36·37-s + 0.871·41-s + 1.14·47-s + 0.142·49-s + 0.175·53-s − 0.177·55-s − 0.898·59-s + 0.768·61-s − 1.12·65-s + 1.79·67-s + 1.43·71-s + 1.15·73-s − 0.0839·77-s − 0.0906·79-s + 1.03·83-s + 0.647·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.140532371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.140532371\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 1.79T + 5T^{2} \) |
| 11 | \( 1 + 0.736T + 11T^{2} \) |
| 13 | \( 1 + 5.06T + 13T^{2} \) |
| 17 | \( 1 - 3.33T + 17T^{2} \) |
| 23 | \( 1 + 7.06T + 23T^{2} \) |
| 29 | \( 1 + 5.27T + 29T^{2} \) |
| 31 | \( 1 + 7.80T + 31T^{2} \) |
| 37 | \( 1 - 8.33T + 37T^{2} \) |
| 41 | \( 1 - 5.58T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 7.84T + 47T^{2} \) |
| 53 | \( 1 - 1.27T + 53T^{2} \) |
| 59 | \( 1 + 6.90T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 14.7T + 67T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 - 9.87T + 73T^{2} \) |
| 79 | \( 1 + 0.805T + 79T^{2} \) |
| 83 | \( 1 - 9.38T + 83T^{2} \) |
| 89 | \( 1 - 4.26T + 89T^{2} \) |
| 97 | \( 1 - 18.9T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.69483541606010408088783795773, −7.16651290043403391744809839586, −6.14511356796167422456840516027, −5.61510946919811668310432851036, −5.11704527273308374420525792952, −4.21071162778156798931014871677, −3.46059325924665562834732104434, −2.23198625118616367925005824542, −2.07004711388864479692031643526, −0.67654009814019598512140254832,
0.67654009814019598512140254832, 2.07004711388864479692031643526, 2.23198625118616367925005824542, 3.46059325924665562834732104434, 4.21071162778156798931014871677, 5.11704527273308374420525792952, 5.61510946919811668310432851036, 6.14511356796167422456840516027, 7.16651290043403391744809839586, 7.69483541606010408088783795773
