L(s) = 1 | − i·2-s + (0.707 − 0.707i)3-s − 4-s + (1.73 − 1.40i)5-s + (−0.707 − 0.707i)6-s + (−3.05 + 3.05i)7-s + i·8-s − 1.00i·9-s + (−1.40 − 1.73i)10-s + 3.00i·11-s + (−0.707 + 0.707i)12-s + 6.48i·13-s + (3.05 + 3.05i)14-s + (0.230 − 2.22i)15-s + 16-s + 6.00·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.408 − 0.408i)3-s − 0.5·4-s + (0.776 − 0.630i)5-s + (−0.288 − 0.288i)6-s + (−1.15 + 1.15i)7-s + 0.353i·8-s − 0.333i·9-s + (−0.445 − 0.548i)10-s + 0.906i·11-s + (−0.204 + 0.204i)12-s + 1.79i·13-s + (0.815 + 0.815i)14-s + (0.0594 − 0.574i)15-s + 0.250·16-s + 1.45·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.544883454\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.544883454\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (-1.73 + 1.40i)T \) |
| 37 | \( 1 + (1.11 + 5.98i)T \) |
good | 7 | \( 1 + (3.05 - 3.05i)T - 7iT^{2} \) |
| 11 | \( 1 - 3.00iT - 11T^{2} \) |
| 13 | \( 1 - 6.48iT - 13T^{2} \) |
| 17 | \( 1 - 6.00T + 17T^{2} \) |
| 19 | \( 1 + (4.44 - 4.44i)T - 19iT^{2} \) |
| 23 | \( 1 - 4.23iT - 23T^{2} \) |
| 29 | \( 1 + (-1.51 - 1.51i)T + 29iT^{2} \) |
| 31 | \( 1 + (-1.07 + 1.07i)T - 31iT^{2} \) |
| 41 | \( 1 + 6.80iT - 41T^{2} \) |
| 43 | \( 1 - 11.7iT - 43T^{2} \) |
| 47 | \( 1 + (-5.31 + 5.31i)T - 47iT^{2} \) |
| 53 | \( 1 + (5.63 + 5.63i)T + 53iT^{2} \) |
| 59 | \( 1 + (4.46 - 4.46i)T - 59iT^{2} \) |
| 61 | \( 1 + (7.40 - 7.40i)T - 61iT^{2} \) |
| 67 | \( 1 + (2.42 + 2.42i)T + 67iT^{2} \) |
| 71 | \( 1 + 9.18T + 71T^{2} \) |
| 73 | \( 1 + (2.01 - 2.01i)T - 73iT^{2} \) |
| 79 | \( 1 + (-6.59 + 6.59i)T - 79iT^{2} \) |
| 83 | \( 1 + (0.481 + 0.481i)T + 83iT^{2} \) |
| 89 | \( 1 + (-4.35 - 4.35i)T + 89iT^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−9.634400556288620141691831509024, −9.264804851673810921722745060999, −8.623506926094815810032664173294, −7.44812575623697625802234551037, −6.31051223600918458713367304890, −5.73471910629282505068926974734, −4.53434575598787702632069528520, −3.44794326173010137851622069656, −2.25400028477110426241138483460, −1.62736463307161982091483534038,
0.65268690708845108270163781088, 3.00140115628791025539623894186, 3.32310207753556734007814357571, 4.67836721213126343255393734244, 5.83282631363217313565188611952, 6.36301147290367867264689506316, 7.30917178428405991909755071066, 8.075274474008902007904963480213, 8.992438310361471852325839722846, 9.969727118774664587670776934941