| L(s) = 1 | − 5.23·2-s + 19.3·4-s + 13.4·5-s + 28.9·7-s − 59.6·8-s − 70.4·10-s − 36.5·11-s − 14.0·13-s − 151.·14-s + 156.·16-s + 34.8·17-s − 115.·19-s + 261.·20-s + 191.·22-s − 175.·23-s + 56.1·25-s + 73.7·26-s + 561.·28-s − 163.·29-s + 141.·31-s − 344.·32-s − 182.·34-s + 389.·35-s − 87.4·37-s + 602.·38-s − 802.·40-s + 517.·41-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 2.42·4-s + 1.20·5-s + 1.56·7-s − 2.63·8-s − 2.22·10-s − 1.00·11-s − 0.300·13-s − 2.89·14-s + 2.45·16-s + 0.496·17-s − 1.39·19-s + 2.91·20-s + 1.85·22-s − 1.58·23-s + 0.449·25-s + 0.556·26-s + 3.78·28-s − 1.04·29-s + 0.820·31-s − 1.90·32-s − 0.919·34-s + 1.88·35-s − 0.388·37-s + 2.57·38-s − 3.17·40-s + 1.97·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 + 5.23T + 8T^{2} \) |
| 5 | \( 1 - 13.4T + 125T^{2} \) |
| 7 | \( 1 - 28.9T + 343T^{2} \) |
| 11 | \( 1 + 36.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 14.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 34.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 115.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 175.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 163.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 141.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 87.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 517.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 221.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 196.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 120.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 586.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 31.9T + 2.26e5T^{2} \) |
| 67 | \( 1 - 411.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 611.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 949.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 383.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 685.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 232.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 758.T + 9.12e5T^{2} \) |
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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
−8.296536761868763176395553430377, −7.898011669260029172256046942895, −7.16874797508255818538419131406, −6.02269802610580400346168198631, −5.57494692848261803243435314703, −4.34442883317296750547893393263, −2.46760301259701386042227622141, −2.09874186845301893645619157291, −1.27309612130708749266264157584, 0,
1.27309612130708749266264157584, 2.09874186845301893645619157291, 2.46760301259701386042227622141, 4.34442883317296750547893393263, 5.57494692848261803243435314703, 6.02269802610580400346168198631, 7.16874797508255818538419131406, 7.898011669260029172256046942895, 8.296536761868763176395553430377
