L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (3.11 − 2.26i)5-s + (0.190 + 0.587i)7-s + (0.309 − 0.951i)8-s + (2.42 + 1.76i)9-s − 3.85·10-s + (2.54 + 2.12i)11-s + (−2 − 1.45i)13-s + (0.190 − 0.587i)14-s + (−0.809 + 0.587i)16-s + (−3.30 + 2.40i)17-s + (−0.927 − 2.85i)18-s + (0.309 − 0.951i)19-s + (3.11 + 2.26i)20-s + ⋯ |
L(s) = 1 | + (−0.572 − 0.415i)2-s + (0.154 + 0.475i)4-s + (1.39 − 1.01i)5-s + (0.0721 + 0.222i)7-s + (0.109 − 0.336i)8-s + (0.809 + 0.587i)9-s − 1.21·10-s + (0.767 + 0.641i)11-s + (−0.554 − 0.403i)13-s + (0.0510 − 0.157i)14-s + (−0.202 + 0.146i)16-s + (−0.802 + 0.583i)17-s + (−0.218 − 0.672i)18-s + (0.0708 − 0.218i)19-s + (0.697 + 0.506i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.32384 - 0.462559i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.32384 - 0.462559i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 11 | \( 1 + (-2.54 - 2.12i)T \) |
| 19 | \( 1 + (-0.309 + 0.951i)T \) |
good | 3 | \( 1 + (-2.42 - 1.76i)T^{2} \) |
| 5 | \( 1 + (-3.11 + 2.26i)T + (1.54 - 4.75i)T^{2} \) |
| 7 | \( 1 + (-0.190 - 0.587i)T + (-5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (2 + 1.45i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (3.30 - 2.40i)T + (5.25 - 16.1i)T^{2} \) |
| 23 | \( 1 - 1.14T + 23T^{2} \) |
| 29 | \( 1 + (-0.381 - 1.17i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-4.61 - 3.35i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (2.85 + 8.78i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.23 + 9.95i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 1.61T + 43T^{2} \) |
| 47 | \( 1 + (3.59 - 11.0i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (1.61 + 1.17i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (3.09 + 9.51i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (8.16 - 5.93i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + (6.85 - 4.97i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (2.61 + 8.05i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-9.70 - 7.05i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (3.11 - 2.26i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + (-6.09 - 4.42i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.79426311943852071813024049604, −10.11026825292495302717116753788, −9.288783424634878777037911378874, −8.769581680394432977253672972722, −7.51145513266564451104704919384, −6.44109758146621749695206065768, −5.21510612809109219303831221242, −4.32111687798624885735925734803, −2.31451592277990591841759347255, −1.44079572516520030742125793663,
1.52038901077431140202826446830, 2.91907774396294469003221831187, 4.57811945031879680456433681279, 6.06223837085238675475656327415, 6.59220376276754030061246984192, 7.32681208293523091011372823298, 8.782721384707271700484804160214, 9.655394544446680966702731259253, 10.06505624439230841087416312497, 11.07633980504441421448943923918