L(s) = 1 | − 2-s + 4-s + 0.761·5-s + 1.23·7-s − 8-s − 0.761·10-s + 4.16·11-s + 2.92·13-s − 1.23·14-s + 16-s + 2.05·17-s + 0.666·19-s + 0.761·20-s − 4.16·22-s + 8.48·23-s − 4.42·25-s − 2.92·26-s + 1.23·28-s − 1.77·29-s + 3.84·31-s − 32-s − 2.05·34-s + 0.943·35-s + 4.44·37-s − 0.666·38-s − 0.761·40-s − 2.52·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.340·5-s + 0.468·7-s − 0.353·8-s − 0.240·10-s + 1.25·11-s + 0.812·13-s − 0.331·14-s + 0.250·16-s + 0.497·17-s + 0.152·19-s + 0.170·20-s − 0.887·22-s + 1.76·23-s − 0.884·25-s − 0.574·26-s + 0.234·28-s − 0.329·29-s + 0.690·31-s − 0.176·32-s − 0.351·34-s + 0.159·35-s + 0.731·37-s − 0.108·38-s − 0.120·40-s − 0.394·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.208885131\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.208885131\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 149 | \( 1 + T \) |
good | 5 | \( 1 - 0.761T + 5T^{2} \) |
| 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 - 4.16T + 11T^{2} \) |
| 13 | \( 1 - 2.92T + 13T^{2} \) |
| 17 | \( 1 - 2.05T + 17T^{2} \) |
| 19 | \( 1 - 0.666T + 19T^{2} \) |
| 23 | \( 1 - 8.48T + 23T^{2} \) |
| 29 | \( 1 + 1.77T + 29T^{2} \) |
| 31 | \( 1 - 3.84T + 31T^{2} \) |
| 37 | \( 1 - 4.44T + 37T^{2} \) |
| 41 | \( 1 + 2.52T + 41T^{2} \) |
| 43 | \( 1 - 5.24T + 43T^{2} \) |
| 47 | \( 1 + 3.71T + 47T^{2} \) |
| 53 | \( 1 - 8.26T + 53T^{2} \) |
| 59 | \( 1 - 9.37T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 + 1.45T + 67T^{2} \) |
| 71 | \( 1 - 6.19T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 + 9.54T + 79T^{2} \) |
| 83 | \( 1 - 5.69T + 83T^{2} \) |
| 89 | \( 1 - 7.72T + 89T^{2} \) |
| 97 | \( 1 + 18.6T + 97T^{2} \) |
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
−7.941884047844664614060794112641, −7.13143682538313879563559161100, −6.58262127864092438515312185797, −5.87268596934110899535629428341, −5.18196459504901058442520511993, −4.17629674446633209245276122965, −3.47169845365501707425517347203, −2.51143329952708325460511240213, −1.47735207762265926396863384043, −0.931836663420192797605230724261,
0.931836663420192797605230724261, 1.47735207762265926396863384043, 2.51143329952708325460511240213, 3.47169845365501707425517347203, 4.17629674446633209245276122965, 5.18196459504901058442520511993, 5.87268596934110899535629428341, 6.58262127864092438515312185797, 7.13143682538313879563559161100, 7.941884047844664614060794112641