| L(s) = 1 | − 2.72·2-s − 3-s + 5.43·4-s + 0.416·5-s + 2.72·6-s − 9.37·8-s + 9-s − 1.13·10-s − 0.0467·11-s − 5.43·12-s − 3.00·13-s − 0.416·15-s + 14.6·16-s + 5.85·17-s − 2.72·18-s − 6.05·19-s + 2.26·20-s + 0.127·22-s + 0.952·23-s + 9.37·24-s − 4.82·25-s + 8.19·26-s − 27-s + 2.00·29-s + 1.13·30-s + 1.80·31-s − 21.3·32-s + ⋯ |
| L(s) = 1 | − 1.92·2-s − 0.577·3-s + 2.71·4-s + 0.186·5-s + 1.11·6-s − 3.31·8-s + 0.333·9-s − 0.358·10-s − 0.0141·11-s − 1.56·12-s − 0.833·13-s − 0.107·15-s + 3.67·16-s + 1.42·17-s − 0.642·18-s − 1.38·19-s + 0.506·20-s + 0.0272·22-s + 0.198·23-s + 1.91·24-s − 0.965·25-s + 1.60·26-s − 0.192·27-s + 0.372·29-s + 0.207·30-s + 0.323·31-s − 3.76·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 2.72T + 2T^{2} \) |
| 5 | \( 1 - 0.416T + 5T^{2} \) |
| 11 | \( 1 + 0.0467T + 11T^{2} \) |
| 13 | \( 1 + 3.00T + 13T^{2} \) |
| 17 | \( 1 - 5.85T + 17T^{2} \) |
| 19 | \( 1 + 6.05T + 19T^{2} \) |
| 23 | \( 1 - 0.952T + 23T^{2} \) |
| 29 | \( 1 - 2.00T + 29T^{2} \) |
| 31 | \( 1 - 1.80T + 31T^{2} \) |
| 37 | \( 1 - 3.23T + 37T^{2} \) |
| 43 | \( 1 + 9.90T + 43T^{2} \) |
| 47 | \( 1 - 9.28T + 47T^{2} \) |
| 53 | \( 1 + 2.46T + 53T^{2} \) |
| 59 | \( 1 - 6.18T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 9.49T + 67T^{2} \) |
| 71 | \( 1 - 4.70T + 71T^{2} \) |
| 73 | \( 1 - 5.34T + 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 - 3.23T + 83T^{2} \) |
| 89 | \( 1 + 4.51T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−7.985037077914359223665581271786, −7.12510794208573223195792154375, −6.59936267252914810259440227855, −5.88765006451532441739172407696, −5.15704093486641830135608581853, −3.87348417339097809381089123272, −2.73410294568973282165583583175, −1.98680973677376612674451772104, −1.02359699599463388018159897127, 0,
1.02359699599463388018159897127, 1.98680973677376612674451772104, 2.73410294568973282165583583175, 3.87348417339097809381089123272, 5.15704093486641830135608581853, 5.88765006451532441739172407696, 6.59936267252914810259440227855, 7.12510794208573223195792154375, 7.985037077914359223665581271786
