| L(s) = 1 | + 0.414·2-s + (1 − 1.41i)3-s − 1.82·4-s + i·5-s + (0.414 − 0.585i)6-s + i·7-s − 1.58·8-s + (−1.00 − 2.82i)9-s + 0.414i·10-s + (1.41 + 3i)11-s + (−1.82 + 2.58i)12-s + 0.414i·14-s + (1.41 + i)15-s + 3·16-s + 6.82·17-s + (−0.414 − 1.17i)18-s + ⋯ |
| L(s) = 1 | + 0.292·2-s + (0.577 − 0.816i)3-s − 0.914·4-s + 0.447i·5-s + (0.169 − 0.239i)6-s + 0.377i·7-s − 0.560·8-s + (−0.333 − 0.942i)9-s + 0.130i·10-s + (0.426 + 0.904i)11-s + (−0.527 + 0.746i)12-s + 0.110i·14-s + (0.365 + 0.258i)15-s + 0.750·16-s + 1.65·17-s + (−0.0976 − 0.276i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.174i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.899704201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.899704201\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1 + 1.41i)T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (-1.41 - 3i)T \) |
| good | 2 | \( 1 - 0.414T + 2T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 6.82T + 17T^{2} \) |
| 19 | \( 1 + 0.828iT - 19T^{2} \) |
| 23 | \( 1 - 0.828iT - 23T^{2} \) |
| 29 | \( 1 - 5.65T + 29T^{2} \) |
| 31 | \( 1 - 0.828T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 7.65T + 41T^{2} \) |
| 43 | \( 1 + 9.65iT - 43T^{2} \) |
| 47 | \( 1 - 10.8iT - 47T^{2} \) |
| 53 | \( 1 - 0.828iT - 53T^{2} \) |
| 59 | \( 1 + 0.343iT - 59T^{2} \) |
| 61 | \( 1 - 5.17iT - 61T^{2} \) |
| 67 | \( 1 + 6.48T + 67T^{2} \) |
| 71 | \( 1 - 0.343iT - 71T^{2} \) |
| 73 | \( 1 + 12iT - 73T^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 - 7.65iT - 89T^{2} \) |
| 97 | \( 1 + 5.17T + 97T^{2} \) |
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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
−9.553960794041287463072524984235, −9.029041641620972855734772532951, −8.003473156311007039527086303799, −7.46911286778279496002039784648, −6.38360144564905115530464059347, −5.62993930176573612868681603999, −4.47484772380230149505242221305, −3.47503266096435616241455344497, −2.57176184614156839015662782998, −1.11057177852346342723139162784,
0.967066258156891092513830663489, 2.96955458023403756790430587966, 3.74026968444689358944525993793, 4.50899027421435645999146800481, 5.35145631153365257257288427174, 6.14218864974407392950114274744, 7.75790433764097897222255131841, 8.285058307626982206580398040412, 9.043376798301655111319620409974, 9.762315885114425115859781450946
