L(s) = 1 | + 2.72·2-s + (−1.44 + 0.948i)3-s + 5.40·4-s + i·5-s + (−3.94 + 2.58i)6-s − i·7-s + 9.27·8-s + (1.20 − 2.74i)9-s + 2.72i·10-s + (3.31 + 0.00239i)11-s + (−7.83 + 5.12i)12-s − 0.172i·13-s − 2.72i·14-s + (−0.948 − 1.44i)15-s + 14.4·16-s − 0.805·17-s + ⋯ |
L(s) = 1 | + 1.92·2-s + (−0.836 + 0.547i)3-s + 2.70·4-s + 0.447i·5-s + (−1.61 + 1.05i)6-s − 0.377i·7-s + 3.27·8-s + (0.400 − 0.916i)9-s + 0.860i·10-s + (0.999 + 0.000720i)11-s + (−2.26 + 1.47i)12-s − 0.0477i·13-s − 0.727i·14-s + (−0.244 − 0.374i)15-s + 3.60·16-s − 0.195·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.486495648\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.486495648\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.44 - 0.948i)T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (-3.31 - 0.00239i)T \) |
good | 2 | \( 1 - 2.72T + 2T^{2} \) |
| 13 | \( 1 + 0.172iT - 13T^{2} \) |
| 17 | \( 1 + 0.805T + 17T^{2} \) |
| 19 | \( 1 - 2.99iT - 19T^{2} \) |
| 23 | \( 1 + 0.277iT - 23T^{2} \) |
| 29 | \( 1 + 7.75T + 29T^{2} \) |
| 31 | \( 1 - 2.70T + 31T^{2} \) |
| 37 | \( 1 + 1.15T + 37T^{2} \) |
| 41 | \( 1 - 2.26T + 41T^{2} \) |
| 43 | \( 1 - 9.92iT - 43T^{2} \) |
| 47 | \( 1 + 11.2iT - 47T^{2} \) |
| 53 | \( 1 - 10.5iT - 53T^{2} \) |
| 59 | \( 1 + 5.52iT - 59T^{2} \) |
| 61 | \( 1 + 4.15iT - 61T^{2} \) |
| 67 | \( 1 + 8.68T + 67T^{2} \) |
| 71 | \( 1 + 14.8iT - 71T^{2} \) |
| 73 | \( 1 + 6.17iT - 73T^{2} \) |
| 79 | \( 1 + 3.40iT - 79T^{2} \) |
| 83 | \( 1 + 18.1T + 83T^{2} \) |
| 89 | \( 1 - 14.7iT - 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−10.34385372169163335811762073379, −9.365003791922970901628852296826, −7.71845951777887798974498601768, −6.84876340927403146176512933442, −6.24814010042647475643069708109, −5.57922695354276326963063836168, −4.57281053847736541664257773224, −3.93205035610717472206658612523, −3.19097469638711316063956870879, −1.64566781931176390776877190866,
1.44228751396451795895876352637, 2.49815859405687862740465394361, 3.86298305864858725435189437994, 4.62583777697747100039274917516, 5.47567004739467595199746289558, 6.05438947835642132709462119017, 6.88190171370984079081138655487, 7.49623964022794943665757656868, 8.789692415318506116437221811126, 10.07333103629596362833296591222