L(s) = 1 | + 1.78·2-s − 7.45·3-s − 4.80·4-s − 5·5-s − 13.3·6-s − 20.9·7-s − 22.8·8-s + 28.5·9-s − 8.94·10-s − 11·11-s + 35.7·12-s + 37.7·13-s − 37.4·14-s + 37.2·15-s − 2.52·16-s + 119.·17-s + 51.0·18-s + 19·19-s + 24.0·20-s + 156.·21-s − 19.6·22-s − 24.1·23-s + 170.·24-s + 25·25-s + 67.5·26-s − 11.4·27-s + 100.·28-s + ⋯ |
L(s) = 1 | + 0.632·2-s − 1.43·3-s − 0.600·4-s − 0.447·5-s − 0.906·6-s − 1.13·7-s − 1.01·8-s + 1.05·9-s − 0.282·10-s − 0.301·11-s + 0.860·12-s + 0.805·13-s − 0.714·14-s + 0.641·15-s − 0.0395·16-s + 1.70·17-s + 0.668·18-s + 0.229·19-s + 0.268·20-s + 1.62·21-s − 0.190·22-s − 0.218·23-s + 1.45·24-s + 0.200·25-s + 0.509·26-s − 0.0812·27-s + 0.678·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 5T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 - 19T \) |
good | 2 | \( 1 - 1.78T + 8T^{2} \) |
| 3 | \( 1 + 7.45T + 27T^{2} \) |
| 7 | \( 1 + 20.9T + 343T^{2} \) |
| 13 | \( 1 - 37.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 119.T + 4.91e3T^{2} \) |
| 23 | \( 1 + 24.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 1.45T + 2.43e4T^{2} \) |
| 31 | \( 1 + 7.62T + 2.97e4T^{2} \) |
| 37 | \( 1 + 402.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 278.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 228.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 409.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 45.2T + 1.48e5T^{2} \) |
| 59 | \( 1 + 203.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 196.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 81.1T + 3.00e5T^{2} \) |
| 71 | \( 1 - 257.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 544.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 77.8T + 4.93e5T^{2} \) |
| 83 | \( 1 + 819.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 992.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.22e3T + 9.12e5T^{2} \) |
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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
−9.298798833202570597629045827856, −8.273986532092053583028399036725, −7.16483217874830264901673598126, −6.18698914056639523456468891873, −5.67653592390697738132752269657, −4.94593903772680317484418465629, −3.81624303651395462935782518732, −3.17303429563704023513312296199, −0.932749171450588690272505439689, 0,
0.932749171450588690272505439689, 3.17303429563704023513312296199, 3.81624303651395462935782518732, 4.94593903772680317484418465629, 5.67653592390697738132752269657, 6.18698914056639523456468891873, 7.16483217874830264901673598126, 8.273986532092053583028399036725, 9.298798833202570597629045827856