L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (−1.85 + 1.25i)5-s + (−3.08 − 3.08i)7-s + (−0.707 − 0.707i)8-s + (−0.426 + 2.19i)10-s − 1.10i·11-s + (−2.55 + 2.55i)13-s − 4.36·14-s − 1.00·16-s + (1.10 − 1.10i)17-s + 7.95i·19-s + (1.25 + 1.85i)20-s + (−0.783 − 0.783i)22-s + (0.707 + 0.707i)23-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.828 + 0.559i)5-s + (−1.16 − 1.16i)7-s + (−0.250 − 0.250i)8-s + (−0.134 + 0.694i)10-s − 0.334i·11-s + (−0.707 + 0.707i)13-s − 1.16·14-s − 0.250·16-s + (0.268 − 0.268i)17-s + 1.82i·19-s + (0.279 + 0.414i)20-s + (−0.167 − 0.167i)22-s + (0.147 + 0.147i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.506i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.861 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.109629918\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.109629918\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.85 - 1.25i)T \) |
| 23 | \( 1 + (-0.707 - 0.707i)T \) |
good | 7 | \( 1 + (3.08 + 3.08i)T + 7iT^{2} \) |
| 11 | \( 1 + 1.10iT - 11T^{2} \) |
| 13 | \( 1 + (2.55 - 2.55i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.10 + 1.10i)T - 17iT^{2} \) |
| 19 | \( 1 - 7.95iT - 19T^{2} \) |
| 29 | \( 1 - 6.30T + 29T^{2} \) |
| 31 | \( 1 - 3.74T + 31T^{2} \) |
| 37 | \( 1 + (-4.56 - 4.56i)T + 37iT^{2} \) |
| 41 | \( 1 - 4.15iT - 41T^{2} \) |
| 43 | \( 1 + (3.87 - 3.87i)T - 43iT^{2} \) |
| 47 | \( 1 + (-6.32 + 6.32i)T - 47iT^{2} \) |
| 53 | \( 1 + (3.25 + 3.25i)T + 53iT^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 + (-0.0854 - 0.0854i)T + 67iT^{2} \) |
| 71 | \( 1 + 2.39iT - 71T^{2} \) |
| 73 | \( 1 + (-2.36 + 2.36i)T - 73iT^{2} \) |
| 79 | \( 1 - 13.4iT - 79T^{2} \) |
| 83 | \( 1 + (-4.53 - 4.53i)T + 83iT^{2} \) |
| 89 | \( 1 - 1.98T + 89T^{2} \) |
| 97 | \( 1 + (-1.85 - 1.85i)T + 97iT^{2} \) |
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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
−9.569221455387480366512771078396, −8.251315664593959839639562708102, −7.56295586030111798901302536405, −6.65083480751575029194598095125, −6.28574094736038507111902324771, −4.90630927913292038454306808832, −4.01624969986099584355426450907, −3.48465425868142486045111410135, −2.64931980461729887030172427442, −1.00304723416484747807939486608,
0.40350981780533589398634918270, 2.58050243402695628489597937407, 3.13274433406898145579527051559, 4.34532210480526050715859390598, 5.03727615981685692801440386215, 5.82644844613959133231179200369, 6.70361859615164949991356322770, 7.39347201983878238882240781037, 8.266569859398221351236583659766, 8.985243095652506363838146198002