| L(s) = 1 | + 1.83·2-s − 4.61·4-s + 1.96·5-s − 1.07·7-s − 23.2·8-s + 3.60·10-s + 30.1·11-s + 3.83·13-s − 1.98·14-s − 5.77·16-s + 74.2·17-s − 12.6·19-s − 9.05·20-s + 55.4·22-s − 184.·23-s − 121.·25-s + 7.05·26-s + 4.96·28-s + 121.·29-s − 62.6·31-s + 175.·32-s + 136.·34-s − 2.11·35-s − 276.·37-s − 23.3·38-s − 45.5·40-s + 172.·41-s + ⋯ |
| L(s) = 1 | + 0.650·2-s − 0.576·4-s + 0.175·5-s − 0.0581·7-s − 1.02·8-s + 0.114·10-s + 0.825·11-s + 0.0818·13-s − 0.0378·14-s − 0.0901·16-s + 1.05·17-s − 0.153·19-s − 0.101·20-s + 0.537·22-s − 1.67·23-s − 0.969·25-s + 0.0532·26-s + 0.0335·28-s + 0.775·29-s − 0.363·31-s + 0.967·32-s + 0.688·34-s − 0.0101·35-s − 1.22·37-s − 0.0997·38-s − 0.179·40-s + 0.656·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 241 | \( 1 + 241T \) |
| good | 2 | \( 1 - 1.83T + 8T^{2} \) |
| 5 | \( 1 - 1.96T + 125T^{2} \) |
| 7 | \( 1 + 1.07T + 343T^{2} \) |
| 11 | \( 1 - 30.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 3.83T + 2.19e3T^{2} \) |
| 17 | \( 1 - 74.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 12.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 184.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 121.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 62.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 276.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 172.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 239.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 346.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 417.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 331.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 440.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 495.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 355.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.04e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 795.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 38.1T + 5.71e5T^{2} \) |
| 89 | \( 1 + 688.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.38e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.351800122403451551435168947226, −7.61644060410642133193376945093, −6.47775503660279226232097942421, −5.85740873070517634245990451208, −5.17697299001886702598148183462, −4.04116547534742683256911076953, −3.72896395394272416750977795289, −2.50294422278027897026337669115, −1.25411357910557944741729118926, 0,
1.25411357910557944741729118926, 2.50294422278027897026337669115, 3.72896395394272416750977795289, 4.04116547534742683256911076953, 5.17697299001886702598148183462, 5.85740873070517634245990451208, 6.47775503660279226232097942421, 7.61644060410642133193376945093, 8.351800122403451551435168947226
