| L(s) = 1 | − 9.48·3-s − 10.6·5-s + 7.29·7-s + 63.0·9-s − 39.6·11-s + 13·13-s + 100.·15-s + 111.·17-s + 70.5·19-s − 69.1·21-s − 3.12·23-s − 11.7·25-s − 341.·27-s + 262.·29-s − 86.5·31-s + 376.·33-s − 77.6·35-s + 61.7·37-s − 123.·39-s − 198.·41-s − 393.·43-s − 670.·45-s − 422.·47-s − 289.·49-s − 1.06e3·51-s + 193.·53-s + 422.·55-s + ⋯ |
| L(s) = 1 | − 1.82·3-s − 0.952·5-s + 0.393·7-s + 2.33·9-s − 1.08·11-s + 0.277·13-s + 1.73·15-s + 1.59·17-s + 0.852·19-s − 0.719·21-s − 0.0283·23-s − 0.0936·25-s − 2.43·27-s + 1.68·29-s − 0.501·31-s + 1.98·33-s − 0.374·35-s + 0.274·37-s − 0.506·39-s − 0.756·41-s − 1.39·43-s − 2.22·45-s − 1.31·47-s − 0.844·49-s − 2.91·51-s + 0.501·53-s + 1.03·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 3 | \( 1 + 9.48T + 27T^{2} \) |
| 5 | \( 1 + 10.6T + 125T^{2} \) |
| 7 | \( 1 - 7.29T + 343T^{2} \) |
| 11 | \( 1 + 39.6T + 1.33e3T^{2} \) |
| 17 | \( 1 - 111.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 70.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 3.12T + 1.21e4T^{2} \) |
| 29 | \( 1 - 262.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 86.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 61.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 198.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 393.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 422.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 193.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 512.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 714.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 498.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 852.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 28.6T + 3.89e5T^{2} \) |
| 79 | \( 1 + 747.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.04e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 278.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.60e3T + 9.12e5T^{2} \) |
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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
−10.46957778634224707190471599866, −9.927211090627066721493208521865, −8.135594666583220762555530305099, −7.51919485480718625561268238202, −6.42986402948959714088163729939, −5.34079503636668852014399344903, −4.81436610691210131371642430019, −3.44168990620366671850498362291, −1.19421114872435626046065655188, 0,
1.19421114872435626046065655188, 3.44168990620366671850498362291, 4.81436610691210131371642430019, 5.34079503636668852014399344903, 6.42986402948959714088163729939, 7.51919485480718625561268238202, 8.135594666583220762555530305099, 9.927211090627066721493208521865, 10.46957778634224707190471599866
