| L(s) = 1 | + 2.45·2-s + 7.08·3-s − 1.98·4-s − 6.77·5-s + 17.3·6-s + 9.04·7-s − 24.4·8-s + 23.1·9-s − 16.6·10-s − 64.1·11-s − 14.0·12-s + 4.88·13-s + 22.1·14-s − 47.9·15-s − 44.2·16-s − 109.·17-s + 56.8·18-s − 40.3·19-s + 13.4·20-s + 64.0·21-s − 157.·22-s − 173.·24-s − 79.1·25-s + 11.9·26-s − 27.2·27-s − 17.9·28-s + 108.·29-s + ⋯ |
| L(s) = 1 | + 0.867·2-s + 1.36·3-s − 0.247·4-s − 0.605·5-s + 1.18·6-s + 0.488·7-s − 1.08·8-s + 0.857·9-s − 0.525·10-s − 1.75·11-s − 0.337·12-s + 0.104·13-s + 0.423·14-s − 0.825·15-s − 0.691·16-s − 1.55·17-s + 0.743·18-s − 0.486·19-s + 0.149·20-s + 0.665·21-s − 1.52·22-s − 1.47·24-s − 0.632·25-s + 0.0904·26-s − 0.194·27-s − 0.120·28-s + 0.693·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 \) |
| good | 2 | \( 1 - 2.45T + 8T^{2} \) |
| 3 | \( 1 - 7.08T + 27T^{2} \) |
| 5 | \( 1 + 6.77T + 125T^{2} \) |
| 7 | \( 1 - 9.04T + 343T^{2} \) |
| 11 | \( 1 + 64.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 4.88T + 2.19e3T^{2} \) |
| 17 | \( 1 + 109.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 40.3T + 6.85e3T^{2} \) |
| 29 | \( 1 - 108.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 335.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 109.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 25.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 214.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 28.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 91.8T + 1.48e5T^{2} \) |
| 59 | \( 1 + 616.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 511.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 668.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.03e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 112.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 446.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 280.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 461.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 231.T + 9.12e5T^{2} \) |
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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
−9.917922814432853625319107631759, −8.764105173368402266063266562273, −8.315995858980140667022612531144, −7.55955590767795017745731642887, −6.17256497550923110316601695204, −4.83905697551908918657731450201, −4.25510385605588428927905446843, −3.06431881183174176119498788561, −2.33386734550693852020987727123, 0,
2.33386734550693852020987727123, 3.06431881183174176119498788561, 4.25510385605588428927905446843, 4.83905697551908918657731450201, 6.17256497550923110316601695204, 7.55955590767795017745731642887, 8.315995858980140667022612531144, 8.764105173368402266063266562273, 9.917922814432853625319107631759
