L(s) = 1 | + (−0.5 − 0.866i)3-s + (0.766 + 0.642i)4-s + (0.173 − 0.984i)5-s + (0.766 − 0.642i)7-s + (−0.499 + 0.866i)9-s + (−0.326 − 1.85i)11-s + (0.173 − 0.984i)12-s + (−0.326 + 0.118i)13-s + (−0.939 + 0.342i)15-s + (0.173 + 0.984i)16-s + (−0.766 + 1.32i)17-s + (0.766 − 0.642i)20-s + (−0.939 − 0.342i)21-s + (−0.939 − 0.342i)25-s + 0.999·27-s + 28-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)3-s + (0.766 + 0.642i)4-s + (0.173 − 0.984i)5-s + (0.766 − 0.642i)7-s + (−0.499 + 0.866i)9-s + (−0.326 − 1.85i)11-s + (0.173 − 0.984i)12-s + (−0.326 + 0.118i)13-s + (−0.939 + 0.342i)15-s + (0.173 + 0.984i)16-s + (−0.766 + 1.32i)17-s + (0.766 − 0.642i)20-s + (−0.939 − 0.342i)21-s + (−0.939 − 0.342i)25-s + 0.999·27-s + 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.054283493\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.054283493\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.173 + 0.984i)T \) |
| 7 | \( 1 + (-0.766 + 0.642i)T \) |
good | 2 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 11 | \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \) |
| 13 | \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \) |
| 17 | \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 31 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 43 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 47 | \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 61 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 67 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 71 | \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \) |
| 83 | \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \) |
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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
−10.55485487757392074310168347954, −8.759463306613252928294366402085, −8.274879574650242376375827917128, −7.71253797810080279762728484700, −6.59622032491886991959151887816, −5.92502540112727206402107814936, −4.92997256565655745417267946506, −3.73852468378739320730951760696, −2.30764194291538261744544394287, −1.16428929235713621715008262618,
2.09101500223484855607635008100, 2.83328640195832994579633222336, 4.56746483893630913332132042221, 5.10398620215298872132173701348, 6.12103384441523442656830370847, 6.94266285841479938052212894731, 7.63349734997524725699808415518, 9.143961544969601982614065544614, 9.846915129591359028400708524225, 10.39503991086906971075984332724