| L(s) = 1 | − 2-s + 2.60·3-s + 4-s + 2.78·5-s − 2.60·6-s − 3.47·7-s − 8-s + 3.78·9-s − 2.78·10-s + 0.290·11-s + 2.60·12-s + 3.47·14-s + 7.25·15-s + 16-s − 7.54·17-s − 3.78·18-s − 19-s + 2.78·20-s − 9.04·21-s − 0.290·22-s − 5.86·23-s − 2.60·24-s + 2.76·25-s + 2.04·27-s − 3.47·28-s + 8.22·29-s − 7.25·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.50·3-s + 0.5·4-s + 1.24·5-s − 1.06·6-s − 1.31·7-s − 0.353·8-s + 1.26·9-s − 0.881·10-s + 0.0876·11-s + 0.752·12-s + 0.927·14-s + 1.87·15-s + 0.250·16-s − 1.83·17-s − 0.892·18-s − 0.229·19-s + 0.622·20-s − 1.97·21-s − 0.0619·22-s − 1.22·23-s − 0.531·24-s + 0.552·25-s + 0.394·27-s − 0.656·28-s + 1.52·29-s − 1.32·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.60T + 3T^{2} \) |
| 5 | \( 1 - 2.78T + 5T^{2} \) |
| 7 | \( 1 + 3.47T + 7T^{2} \) |
| 11 | \( 1 - 0.290T + 11T^{2} \) |
| 17 | \( 1 + 7.54T + 17T^{2} \) |
| 23 | \( 1 + 5.86T + 23T^{2} \) |
| 29 | \( 1 - 8.22T + 29T^{2} \) |
| 31 | \( 1 + 5.41T + 31T^{2} \) |
| 37 | \( 1 + 1.37T + 37T^{2} \) |
| 41 | \( 1 + 4.80T + 41T^{2} \) |
| 43 | \( 1 + 8.94T + 43T^{2} \) |
| 47 | \( 1 - 6.19T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 + 5.57T + 59T^{2} \) |
| 61 | \( 1 - 1.41T + 61T^{2} \) |
| 67 | \( 1 - 8.72T + 67T^{2} \) |
| 71 | \( 1 + 1.27T + 71T^{2} \) |
| 73 | \( 1 + 4.36T + 73T^{2} \) |
| 79 | \( 1 + 9.88T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 - 5.53T + 97T^{2} \) |
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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
−7.890134438973891795823652963652, −6.90378815692188580693531776079, −6.50281097949726722665941883452, −5.88991875809571780974015586633, −4.65361986581578525789071067840, −3.66481619932838977726156245313, −2.94182321781733616255776363607, −2.22777389702264530635845520277, −1.72015889904581436138302105625, 0,
1.72015889904581436138302105625, 2.22777389702264530635845520277, 2.94182321781733616255776363607, 3.66481619932838977726156245313, 4.65361986581578525789071067840, 5.88991875809571780974015586633, 6.50281097949726722665941883452, 6.90378815692188580693531776079, 7.890134438973891795823652963652
