| L(s) = 1 | + 5.13·2-s + 18.3·4-s − 5·5-s − 7.03·7-s + 52.9·8-s − 25.6·10-s − 47.2·11-s − 7.36·13-s − 36.0·14-s + 125.·16-s + 15.0·17-s − 117.·19-s − 91.6·20-s − 242.·22-s − 97.0·23-s + 25·25-s − 37.8·26-s − 128.·28-s + 29·29-s + 112.·31-s + 218.·32-s + 77.2·34-s + 35.1·35-s − 128.·37-s − 604.·38-s − 264.·40-s − 432.·41-s + ⋯ |
| L(s) = 1 | + 1.81·2-s + 2.29·4-s − 0.447·5-s − 0.379·7-s + 2.34·8-s − 0.811·10-s − 1.29·11-s − 0.157·13-s − 0.688·14-s + 1.95·16-s + 0.214·17-s − 1.42·19-s − 1.02·20-s − 2.35·22-s − 0.880·23-s + 0.200·25-s − 0.285·26-s − 0.869·28-s + 0.185·29-s + 0.650·31-s + 1.20·32-s + 0.389·34-s + 0.169·35-s − 0.572·37-s − 2.57·38-s − 1.04·40-s − 1.64·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 - 5.13T + 8T^{2} \) |
| 7 | \( 1 + 7.03T + 343T^{2} \) |
| 11 | \( 1 + 47.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 7.36T + 2.19e3T^{2} \) |
| 17 | \( 1 - 15.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 117.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 97.0T + 1.21e4T^{2} \) |
| 31 | \( 1 - 112.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 128.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 432.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 227.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 280.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 393.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 226.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 65.2T + 2.26e5T^{2} \) |
| 67 | \( 1 + 476.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 65.8T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.06e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 923.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 34.9T + 5.71e5T^{2} \) |
| 89 | \( 1 - 404.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.44e3T + 9.12e5T^{2} \) |
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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
−8.605463175234847324574605541549, −7.80409287337618403136827763733, −6.90383884024154204570688059805, −6.19559192333955811025886012537, −5.31688518705410480049785035120, −4.59581005400666137125924514196, −3.73788854248368094347821711272, −2.87657558300825658518994427597, −2.00279495293136063602715170222, 0,
2.00279495293136063602715170222, 2.87657558300825658518994427597, 3.73788854248368094347821711272, 4.59581005400666137125924514196, 5.31688518705410480049785035120, 6.19559192333955811025886012537, 6.90383884024154204570688059805, 7.80409287337618403136827763733, 8.605463175234847324574605541549
