| L(s) = 1 | + 5.59·2-s − 4.33·3-s + 23.3·4-s − 9.25·5-s − 24.2·6-s + 18.6·7-s + 85.8·8-s − 8.22·9-s − 51.8·10-s − 101.·12-s + 13·13-s + 104.·14-s + 40.1·15-s + 293.·16-s − 86.0·17-s − 46.0·18-s + 49.1·19-s − 215.·20-s − 80.8·21-s + 122.·23-s − 371.·24-s − 39.3·25-s + 72.7·26-s + 152.·27-s + 435.·28-s + 121.·29-s + 224.·30-s + ⋯ |
| L(s) = 1 | + 1.97·2-s − 0.833·3-s + 2.91·4-s − 0.827·5-s − 1.65·6-s + 1.00·7-s + 3.79·8-s − 0.304·9-s − 1.63·10-s − 2.43·12-s + 0.277·13-s + 1.99·14-s + 0.690·15-s + 4.58·16-s − 1.22·17-s − 0.602·18-s + 0.592·19-s − 2.41·20-s − 0.840·21-s + 1.11·23-s − 3.16·24-s − 0.314·25-s + 0.548·26-s + 1.08·27-s + 2.93·28-s + 0.779·29-s + 1.36·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.534586604\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.534586604\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 2 | \( 1 - 5.59T + 8T^{2} \) |
| 3 | \( 1 + 4.33T + 27T^{2} \) |
| 5 | \( 1 + 9.25T + 125T^{2} \) |
| 7 | \( 1 - 18.6T + 343T^{2} \) |
| 17 | \( 1 + 86.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 49.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 122.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 121.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 94.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 69.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 447.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 271.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 388.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 35.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + 280.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 362.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 654.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 773.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 254.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 691.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 140.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 834.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 235.T + 9.12e5T^{2} \) |
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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
−8.888075121310850155710270338131, −7.75025181117882038405781860608, −7.24560761801370470601282875649, −6.23484316533111737582724328670, −5.66418144507644764433143113379, −4.69138055372281858092596251985, −4.42398184993679932508356842566, −3.29178379702402491635381295593, −2.31462440431108277298180467795, −0.993858735766769574269381730176,
0.993858735766769574269381730176, 2.31462440431108277298180467795, 3.29178379702402491635381295593, 4.42398184993679932508356842566, 4.69138055372281858092596251985, 5.66418144507644764433143113379, 6.23484316533111737582724328670, 7.24560761801370470601282875649, 7.75025181117882038405781860608, 8.888075121310850155710270338131
