| L(s) = 1 | + i·2-s − 2.59i·3-s − 4-s − 3.53i·5-s + 2.59·6-s − i·8-s − 3.73·9-s + 3.53·10-s + (−2.29 − 2.39i)11-s + 2.59i·12-s − 5.01·13-s − 9.18·15-s + 16-s + 3.89·17-s − 3.73i·18-s + 4.65·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 1.49i·3-s − 0.5·4-s − 1.58i·5-s + 1.05·6-s − 0.353i·8-s − 1.24·9-s + 1.11·10-s + (−0.691 − 0.722i)11-s + 0.749i·12-s − 1.39·13-s − 2.37·15-s + 0.250·16-s + 0.944·17-s − 0.880i·18-s + 1.06·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0502i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0502i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9476000971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9476000971\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (2.29 + 2.39i)T \) |
| good | 3 | \( 1 + 2.59iT - 3T^{2} \) |
| 5 | \( 1 + 3.53iT - 5T^{2} \) |
| 13 | \( 1 + 5.01T + 13T^{2} \) |
| 17 | \( 1 - 3.89T + 17T^{2} \) |
| 19 | \( 1 - 4.65T + 19T^{2} \) |
| 23 | \( 1 + 1.55T + 23T^{2} \) |
| 29 | \( 1 - 0.100iT - 29T^{2} \) |
| 31 | \( 1 - 0.279iT - 31T^{2} \) |
| 37 | \( 1 - 0.705T + 37T^{2} \) |
| 41 | \( 1 - 2.94T + 41T^{2} \) |
| 43 | \( 1 - 9.03iT - 43T^{2} \) |
| 47 | \( 1 - 6.56iT - 47T^{2} \) |
| 53 | \( 1 - 5.54T + 53T^{2} \) |
| 59 | \( 1 + 14.5iT - 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 - 3.98T + 67T^{2} \) |
| 71 | \( 1 + 8.45T + 71T^{2} \) |
| 73 | \( 1 + 3.89T + 73T^{2} \) |
| 79 | \( 1 - 7.69iT - 79T^{2} \) |
| 83 | \( 1 - 2.87T + 83T^{2} \) |
| 89 | \( 1 + 5.94iT - 89T^{2} \) |
| 97 | \( 1 + 16.7iT - 97T^{2} \) |
| 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
−9.263616118568984108286181806653, −8.211856595925185889247065237845, −7.85776710828899106701703357527, −7.20699510042891877083404191714, −5.99792331784694475339065211980, −5.39681945654499116933437918992, −4.59264302345442531902204884258, −2.92213094022793112183354559984, −1.44457951282400865636534871332, −0.43209907779437602848677662617,
2.38979279419576225865530518234, 3.09461976206237607063316299291, 3.94012324488804225165265795450, 4.95915309366614724795230050140, 5.67659681490601860957447351512, 7.17091831791889585526904377850, 7.70241553300547447453247704289, 9.147567704732779446519939408658, 9.861129913569326348968575400747, 10.32781326752039247684288265450
