| L(s) = 1 | − 4·2-s + 9.07·3-s + 16·4-s + 62.0·5-s − 36.2·6-s − 113.·7-s − 64·8-s − 160.·9-s − 248.·10-s + 263.·11-s + 145.·12-s − 273.·13-s + 453.·14-s + 562.·15-s + 256·16-s − 857.·17-s + 642.·18-s + 1.13e3·19-s + 992.·20-s − 1.02e3·21-s − 1.05e3·22-s + 2.42e3·23-s − 580.·24-s + 721.·25-s + 1.09e3·26-s − 3.66e3·27-s − 1.81e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.581·3-s + 0.5·4-s + 1.10·5-s − 0.411·6-s − 0.874·7-s − 0.353·8-s − 0.661·9-s − 0.784·10-s + 0.656·11-s + 0.290·12-s − 0.448·13-s + 0.618·14-s + 0.645·15-s + 0.250·16-s − 0.719·17-s + 0.467·18-s + 0.720·19-s + 0.554·20-s − 0.509·21-s − 0.463·22-s + 0.956·23-s − 0.205·24-s + 0.230·25-s + 0.317·26-s − 0.966·27-s − 0.437·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 269 | \( 1 + 7.23e4T \) |
| good | 3 | \( 1 - 9.07T + 243T^{2} \) |
| 5 | \( 1 - 62.0T + 3.12e3T^{2} \) |
| 7 | \( 1 + 113.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 263.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 273.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 857.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.13e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.42e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.69e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.92e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 757.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 199.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.12e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.82e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.96e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.45e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.38e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.72e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.75e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.82e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.95e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 559.T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.00e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.19e5T + 8.58e9T^{2} \) |
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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
−9.385443298159200445221790767497, −9.053002483220135473973233796849, −7.993385624632637651359826785639, −6.77780598692035858434833035937, −6.21809440976821910443587537024, −5.05268355107760886811231791610, −3.33770336010769079915134167175, −2.57176511059207811057052481746, −1.45450421511945341471440460090, 0,
1.45450421511945341471440460090, 2.57176511059207811057052481746, 3.33770336010769079915134167175, 5.05268355107760886811231791610, 6.21809440976821910443587537024, 6.77780598692035858434833035937, 7.993385624632637651359826785639, 9.053002483220135473973233796849, 9.385443298159200445221790767497
