| L(s) = 1 | − 2.68·2-s + 5.18·4-s + 1.17·5-s + 7-s − 8.54·8-s − 3.13·10-s − 1.68·13-s − 2.68·14-s + 12.5·16-s − 5.59·17-s + 4.35·19-s + 6.07·20-s − 0.119·23-s − 3.63·25-s + 4.51·26-s + 5.18·28-s − 4.39·29-s − 6.31·31-s − 16.5·32-s + 15.0·34-s + 1.17·35-s + 5.85·37-s − 11.6·38-s − 9.99·40-s − 11.0·41-s + 5.97·43-s + 0.320·46-s + ⋯ |
| L(s) = 1 | − 1.89·2-s + 2.59·4-s + 0.523·5-s + 0.377·7-s − 3.02·8-s − 0.992·10-s − 0.466·13-s − 0.716·14-s + 3.13·16-s − 1.35·17-s + 0.998·19-s + 1.35·20-s − 0.0249·23-s − 0.726·25-s + 0.884·26-s + 0.980·28-s − 0.816·29-s − 1.13·31-s − 2.91·32-s + 2.57·34-s + 0.197·35-s + 0.962·37-s − 1.89·38-s − 1.58·40-s − 1.72·41-s + 0.910·43-s + 0.0472·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7138495111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7138495111\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.68T + 2T^{2} \) |
| 5 | \( 1 - 1.17T + 5T^{2} \) |
| 13 | \( 1 + 1.68T + 13T^{2} \) |
| 17 | \( 1 + 5.59T + 17T^{2} \) |
| 19 | \( 1 - 4.35T + 19T^{2} \) |
| 23 | \( 1 + 0.119T + 23T^{2} \) |
| 29 | \( 1 + 4.39T + 29T^{2} \) |
| 31 | \( 1 + 6.31T + 31T^{2} \) |
| 37 | \( 1 - 5.85T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 - 5.97T + 43T^{2} \) |
| 47 | \( 1 - 12.5T + 47T^{2} \) |
| 53 | \( 1 - 0.417T + 53T^{2} \) |
| 59 | \( 1 + 0.723T + 59T^{2} \) |
| 61 | \( 1 + 8.85T + 61T^{2} \) |
| 67 | \( 1 - 8.78T + 67T^{2} \) |
| 71 | \( 1 - 3.21T + 71T^{2} \) |
| 73 | \( 1 + 14.5T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 + 3.45T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 + 5.99T + 97T^{2} \) |
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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
−7.932239396113975755640796128333, −7.33305129302908929280077061113, −6.88961079138409812535166729392, −5.96881123436764030228157277874, −5.43998676528858655144039606242, −4.25527758432103998143568829469, −3.09345949447602338429569988076, −2.17631374487704363857296398425, −1.72629358672953729158652966838, −0.55237059145286590960719476806,
0.55237059145286590960719476806, 1.72629358672953729158652966838, 2.17631374487704363857296398425, 3.09345949447602338429569988076, 4.25527758432103998143568829469, 5.43998676528858655144039606242, 5.96881123436764030228157277874, 6.88961079138409812535166729392, 7.33305129302908929280077061113, 7.932239396113975755640796128333
