L(s) = 1 | + (1.69 + 0.340i)3-s + 3.78i·5-s + 4.37·7-s + (2.76 + 1.15i)9-s + 2.08i·11-s − 5.19i·13-s + (−1.28 + 6.43i)15-s − 3.88i·17-s + (−1.35 + 4.14i)19-s + (7.43 + 1.48i)21-s − 2.20i·23-s − 9.35·25-s + (4.30 + 2.90i)27-s − 5.53·29-s − 6.99i·31-s + ⋯ |
L(s) = 1 | + (0.980 + 0.196i)3-s + 1.69i·5-s + 1.65·7-s + (0.922 + 0.385i)9-s + 0.629i·11-s − 1.44i·13-s + (−0.332 + 1.66i)15-s − 0.941i·17-s + (−0.311 + 0.950i)19-s + (1.62 + 0.325i)21-s − 0.459i·23-s − 1.87·25-s + (0.829 + 0.559i)27-s − 1.02·29-s − 1.25i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.26019 + 1.31812i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.26019 + 1.31812i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.69 - 0.340i)T \) |
| 19 | \( 1 + (1.35 - 4.14i)T \) |
good | 5 | \( 1 - 3.78iT - 5T^{2} \) |
| 7 | \( 1 - 4.37T + 7T^{2} \) |
| 11 | \( 1 - 2.08iT - 11T^{2} \) |
| 13 | \( 1 + 5.19iT - 13T^{2} \) |
| 17 | \( 1 + 3.88iT - 17T^{2} \) |
| 23 | \( 1 + 2.20iT - 23T^{2} \) |
| 29 | \( 1 + 5.53T + 29T^{2} \) |
| 31 | \( 1 + 6.99iT - 31T^{2} \) |
| 37 | \( 1 + 4.69iT - 37T^{2} \) |
| 41 | \( 1 + 2.14T + 41T^{2} \) |
| 43 | \( 1 + 3.56T + 43T^{2} \) |
| 47 | \( 1 - 7.44iT - 47T^{2} \) |
| 53 | \( 1 + 8.11T + 53T^{2} \) |
| 59 | \( 1 - 3.79T + 59T^{2} \) |
| 61 | \( 1 + 3.12T + 61T^{2} \) |
| 67 | \( 1 - 8.92iT - 67T^{2} \) |
| 71 | \( 1 - 9.65T + 71T^{2} \) |
| 73 | \( 1 + 7.13T + 73T^{2} \) |
| 79 | \( 1 + 11.8iT - 79T^{2} \) |
| 83 | \( 1 + 2.29iT - 83T^{2} \) |
| 89 | \( 1 - 8.17T + 89T^{2} \) |
| 97 | \( 1 + 2.33iT - 97T^{2} \) |
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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
−10.29954053500260339886320638587, −9.544383848710907813253218497888, −8.294502640131611694717506227432, −7.65297011419188090144884120151, −7.27991967957922719041932027386, −5.92126461372233641747569943648, −4.79229466560077540483748429808, −3.74039746646825112235922150443, −2.72619175092534436481959289676, −1.90205041540066398451972947889,
1.35878077448480257066216147225, 1.92270443361847841231901226351, 3.78116462319607553190001284290, 4.61476396947519780978918093295, 5.22811156430859768749538754489, 6.66567808177040560864274021396, 7.80253222165060294097840148696, 8.478705313627471937025024090051, 8.812132148152689402413606920882, 9.556479933499574964433581870387