| L(s) = 1 | + 0.428·3-s − i·5-s − 2.67i·7-s − 2.81·9-s + 5.10i·11-s + (−1.57 − 3.24i)13-s − 0.428i·15-s − 5.34·17-s − 6.24i·19-s − 1.14i·21-s − 2.42·23-s − 25-s − 2.48·27-s + 2.67·29-s + 0.244i·31-s + ⋯ |
| L(s) = 1 | + 0.247·3-s − 0.447i·5-s − 1.01i·7-s − 0.938·9-s + 1.53i·11-s + (−0.435 − 0.899i)13-s − 0.110i·15-s − 1.29·17-s − 1.43i·19-s − 0.249i·21-s − 0.506·23-s − 0.200·25-s − 0.479·27-s + 0.496·29-s + 0.0439i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 + 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.899 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6597128116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6597128116\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 + (1.57 + 3.24i)T \) |
| good | 3 | \( 1 - 0.428T + 3T^{2} \) |
| 7 | \( 1 + 2.67iT - 7T^{2} \) |
| 11 | \( 1 - 5.10iT - 11T^{2} \) |
| 17 | \( 1 + 5.34T + 17T^{2} \) |
| 19 | \( 1 + 6.24iT - 19T^{2} \) |
| 23 | \( 1 + 2.42T + 23T^{2} \) |
| 29 | \( 1 - 2.67T + 29T^{2} \) |
| 31 | \( 1 - 0.244iT - 31T^{2} \) |
| 37 | \( 1 + 3.32iT - 37T^{2} \) |
| 41 | \( 1 - 6.48iT - 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 - 2.67iT - 47T^{2} \) |
| 53 | \( 1 + 4.20T + 53T^{2} \) |
| 59 | \( 1 + 0.899iT - 59T^{2} \) |
| 61 | \( 1 + 5.81T + 61T^{2} \) |
| 67 | \( 1 + 2.18iT - 67T^{2} \) |
| 71 | \( 1 + 6.24iT - 71T^{2} \) |
| 73 | \( 1 + 10.9iT - 73T^{2} \) |
| 79 | \( 1 - 3.63T + 79T^{2} \) |
| 83 | \( 1 + 9.81iT - 83T^{2} \) |
| 89 | \( 1 + 7.63iT - 89T^{2} \) |
| 97 | \( 1 + 11.3iT - 97T^{2} \) |
| 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
−9.550713641150762731322167480342, −8.784627572057289592542906314445, −7.86295707742877476342305059847, −7.16749163170544985123488002894, −6.28519511404368917601073257696, −4.89734123580705336144739721991, −4.50627397812171253386922824444, −3.12887841886027475752014459923, −2.02225992670686468398078587680, −0.26313977640713853560063686577,
2.04316796565009037816323529089, 2.93956991475636749735735694695, 3.91433464490982140398520088707, 5.31456422491974525304038702641, 6.05001883328803098888892516872, 6.72347575888732880288727641428, 8.131513976474306528084804929407, 8.562786682376566164998515797370, 9.264611346529771428889267618309, 10.28367328340485663729084088334
