L(s) = 1 | + (1.32 + 0.483i)2-s + (0.245 + 1.39i)3-s + (0.766 + 0.642i)4-s + (−0.766 + 0.642i)5-s + (−0.347 + 1.96i)6-s + (−0.939 + 0.342i)9-s + (−1.32 + 0.483i)10-s + (−0.707 + 1.22i)12-s + (−0.245 + 1.39i)13-s + (−1.08 − 0.909i)15-s + (−0.173 − 0.984i)16-s − 1.41·18-s − 0.999·20-s + (0.173 − 0.984i)25-s + (−1 + 1.73i)26-s + ⋯ |
L(s) = 1 | + (1.32 + 0.483i)2-s + (0.245 + 1.39i)3-s + (0.766 + 0.642i)4-s + (−0.766 + 0.642i)5-s + (−0.347 + 1.96i)6-s + (−0.939 + 0.342i)9-s + (−1.32 + 0.483i)10-s + (−0.707 + 1.22i)12-s + (−0.245 + 1.39i)13-s + (−1.08 − 0.909i)15-s + (−0.173 − 0.984i)16-s − 1.41·18-s − 0.999·20-s + (0.173 − 0.984i)25-s + (−1 + 1.73i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.021686396\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.021686396\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.766 - 0.642i)T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-1.32 - 0.483i)T + (0.766 + 0.642i)T^{2} \) |
| 3 | \( 1 + (-0.245 - 1.39i)T + (-0.939 + 0.342i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.245 - 1.39i)T + (-0.939 - 0.342i)T^{2} \) |
| 17 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 23 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - 1.41T + T^{2} \) |
| 41 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 43 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 53 | \( 1 + (-1.08 - 0.909i)T + (0.173 + 0.984i)T^{2} \) |
| 59 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 61 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 67 | \( 1 + (-1.32 + 0.483i)T + (0.766 - 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 73 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 79 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 97 | \( 1 + (1.32 + 0.483i)T + (0.766 + 0.642i)T^{2} \) |
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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
−9.728282869934875049367839123695, −9.167823846669985766190971057867, −8.121133604253418552928820397039, −7.11029764811145618448815560641, −6.52884732589837190647599535818, −5.51012849723697429511291276726, −4.56631453167366523835540162179, −4.17334276673113207413300804850, −3.49443910345540789027740995870, −2.57663511492882243031011014158,
1.01337289331048794194138173222, 2.31253465616838711795928023632, 3.13234520492609238663944104419, 4.07753683118058383744688836439, 5.02716619181511411408551203972, 5.75152015933539729279390319539, 6.64744606856370262430806136757, 7.64676356087994395937613167583, 8.079055493071417520499133128539, 8.852190799626281882771096982399