| L(s) = 1 | + 2-s + 4-s + 3.68·5-s + 1.75·7-s + 8-s + 3.68·10-s − 3.04·11-s − 2.39·13-s + 1.75·14-s + 16-s + 3.38·17-s + 3.51·19-s + 3.68·20-s − 3.04·22-s + 8.61·25-s − 2.39·26-s + 1.75·28-s − 3.48·29-s − 1.57·31-s + 32-s + 3.38·34-s + 6.48·35-s + 10.4·37-s + 3.51·38-s + 3.68·40-s + 3.17·41-s − 11.7·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.65·5-s + 0.664·7-s + 0.353·8-s + 1.16·10-s − 0.918·11-s − 0.664·13-s + 0.469·14-s + 0.250·16-s + 0.821·17-s + 0.806·19-s + 0.825·20-s − 0.649·22-s + 1.72·25-s − 0.469·26-s + 0.332·28-s − 0.647·29-s − 0.283·31-s + 0.176·32-s + 0.581·34-s + 1.09·35-s + 1.71·37-s + 0.570·38-s + 0.583·40-s + 0.496·41-s − 1.79·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.280395738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.280395738\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 3.68T + 5T^{2} \) |
| 7 | \( 1 - 1.75T + 7T^{2} \) |
| 11 | \( 1 + 3.04T + 11T^{2} \) |
| 13 | \( 1 + 2.39T + 13T^{2} \) |
| 17 | \( 1 - 3.38T + 17T^{2} \) |
| 19 | \( 1 - 3.51T + 19T^{2} \) |
| 29 | \( 1 + 3.48T + 29T^{2} \) |
| 31 | \( 1 + 1.57T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 3.17T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 - 9.10T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 - 0.604T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 + 7.90T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 17.8T + 83T^{2} \) |
| 89 | \( 1 - 8.55T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−7.61516805970364593340145080563, −6.93246500399748518927375366384, −6.07081834459327102482592578492, −5.43933733530094505848196629559, −5.23311971199526003394134069325, −4.41441875012937623446178938855, −3.29968157873570227975129589671, −2.52173745549725147949305899376, −1.98317003610212817397669925261, −1.02947730847134953534268135144,
1.02947730847134953534268135144, 1.98317003610212817397669925261, 2.52173745549725147949305899376, 3.29968157873570227975129589671, 4.41441875012937623446178938855, 5.23311971199526003394134069325, 5.43933733530094505848196629559, 6.07081834459327102482592578492, 6.93246500399748518927375366384, 7.61516805970364593340145080563
