L(s) = 1 | + (0.615 − 1.06i)2-s + (0.669 − 0.743i)3-s + (−0.258 − 0.447i)4-s + (0.882 + 1.52i)5-s + (−0.380 − 1.17i)6-s + 0.595·8-s + (−0.104 − 0.994i)9-s + 2.17·10-s + (−0.438 + 0.759i)11-s + (−0.504 − 0.107i)12-s + (1.72 + 0.367i)15-s + (0.624 − 1.08i)16-s − 1.87·17-s + (−1.12 − 0.500i)18-s + (0.455 − 0.789i)20-s + ⋯ |
L(s) = 1 | + (0.615 − 1.06i)2-s + (0.669 − 0.743i)3-s + (−0.258 − 0.447i)4-s + (0.882 + 1.52i)5-s + (−0.380 − 1.17i)6-s + 0.595·8-s + (−0.104 − 0.994i)9-s + 2.17·10-s + (−0.438 + 0.759i)11-s + (−0.504 − 0.107i)12-s + (1.72 + 0.367i)15-s + (0.624 − 1.08i)16-s − 1.87·17-s + (−1.12 − 0.500i)18-s + (0.455 − 0.789i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.348537333\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.348537333\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.669 + 0.743i)T \) |
| 239 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + (-0.615 + 1.06i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (-0.882 - 1.52i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.438 - 0.759i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + 1.87T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.719 + 1.24i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.374 - 0.648i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + 1.61T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.961 - 1.66i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−9.425306115610237941140024290235, −8.310544724810099415954396907950, −7.40684336655687804531984093870, −6.76603293962416093889558857966, −6.22374478681585136111586697701, −4.89219376475542144459084135068, −3.88712924140766114705937483677, −2.91590647046062582673683102804, −2.36855743433133820284109334828, −1.82403661199129300891115171628,
1.58506419278609456547249467214, 2.75040765097142924306720339971, 4.29516351890170091740020258394, 4.58207564936077763430385392647, 5.44464369071671900942615112725, 5.95788632012803540745673108641, 6.97368187142172889708061534355, 8.021243174066405914221048813284, 8.747484627881923774419016206702, 8.951221113099828380058817863249