| L(s) = 1 | + 4.27·2-s − 11.5·3-s − 13.7·4-s + 12.1·5-s − 49.5·6-s − 81.5·7-s − 195.·8-s − 108.·9-s + 51.8·10-s − 669.·11-s + 158.·12-s + 731.·13-s − 349.·14-s − 140.·15-s − 397.·16-s + 2.13e3·17-s − 465.·18-s − 1.87e3·19-s − 166.·20-s + 945.·21-s − 2.86e3·22-s + 1.80e3·23-s + 2.26e3·24-s − 2.97e3·25-s + 3.12e3·26-s + 4.07e3·27-s + 1.11e3·28-s + ⋯ |
| L(s) = 1 | + 0.756·2-s − 0.743·3-s − 0.428·4-s + 0.216·5-s − 0.561·6-s − 0.629·7-s − 1.07·8-s − 0.447·9-s + 0.163·10-s − 1.66·11-s + 0.318·12-s + 1.20·13-s − 0.475·14-s − 0.161·15-s − 0.388·16-s + 1.79·17-s − 0.338·18-s − 1.19·19-s − 0.0928·20-s + 0.467·21-s − 1.26·22-s + 0.711·23-s + 0.802·24-s − 0.953·25-s + 0.907·26-s + 1.07·27-s + 0.269·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + 1.36e3T \) |
| good | 2 | \( 1 - 4.27T + 32T^{2} \) |
| 3 | \( 1 + 11.5T + 243T^{2} \) |
| 5 | \( 1 - 12.1T + 3.12e3T^{2} \) |
| 7 | \( 1 + 81.5T + 1.68e4T^{2} \) |
| 11 | \( 1 + 669.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 731.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.13e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.87e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.80e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.02e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.02e3T + 2.86e7T^{2} \) |
| 41 | \( 1 - 3.98e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.00e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.49e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.75e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.99e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.44e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.33e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.89e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.21e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.35e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.64e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.83e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.38e4T + 8.58e9T^{2} \) |
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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
−14.63973287789981280642699527583, −13.30664215032589795307078156402, −12.69403254874410428381566576544, −11.22830907008523749416880561018, −9.919310018509821805305641330266, −8.272311634083730274581349465153, −6.07400461097015585447830508583, −5.27438630646345199628050475351, −3.31757310784119630656231181565, 0,
3.31757310784119630656231181565, 5.27438630646345199628050475351, 6.07400461097015585447830508583, 8.272311634083730274581349465153, 9.919310018509821805305641330266, 11.22830907008523749416880561018, 12.69403254874410428381566576544, 13.30664215032589795307078156402, 14.63973287789981280642699527583
