| L(s) = 1 | + 3.28·2-s + 2.79·4-s − 9.57·5-s + 10.7·7-s − 17.1·8-s − 31.4·10-s + 3.26·11-s + 18.0·13-s + 35.2·14-s − 78.5·16-s + 102.·17-s − 11.5·19-s − 26.7·20-s + 10.7·22-s + 70.2·23-s − 33.2·25-s + 59.2·26-s + 29.9·28-s − 228.·29-s − 51.9·31-s − 121.·32-s + 337.·34-s − 102.·35-s + 83.4·37-s − 37.9·38-s + 163.·40-s + 209.·41-s + ⋯ |
| L(s) = 1 | + 1.16·2-s + 0.348·4-s − 0.856·5-s + 0.578·7-s − 0.756·8-s − 0.995·10-s + 0.0894·11-s + 0.384·13-s + 0.672·14-s − 1.22·16-s + 1.46·17-s − 0.139·19-s − 0.298·20-s + 0.103·22-s + 0.637·23-s − 0.265·25-s + 0.446·26-s + 0.202·28-s − 1.46·29-s − 0.300·31-s − 0.669·32-s + 1.70·34-s − 0.496·35-s + 0.370·37-s − 0.161·38-s + 0.647·40-s + 0.799·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 - 3.28T + 8T^{2} \) |
| 5 | \( 1 + 9.57T + 125T^{2} \) |
| 7 | \( 1 - 10.7T + 343T^{2} \) |
| 11 | \( 1 - 3.26T + 1.33e3T^{2} \) |
| 13 | \( 1 - 18.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 102.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 11.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 70.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 228.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 51.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 83.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 209.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 117.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 37.2T + 1.03e5T^{2} \) |
| 53 | \( 1 + 108.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 233.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 685.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 5.35T + 3.00e5T^{2} \) |
| 71 | \( 1 + 286.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 41.9T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.04e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 298.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 355.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.62e3T + 9.12e5T^{2} \) |
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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
−8.146627105383223741373878570836, −7.58837948844956894517289821597, −6.63562521225087766578597788110, −5.62109601000354242739842946964, −5.16809613374296002304960076102, −4.08853466847820432440692271390, −3.69905703579698346512439947402, −2.73247296296942249451874816046, −1.33822684210705440011135759698, 0,
1.33822684210705440011135759698, 2.73247296296942249451874816046, 3.69905703579698346512439947402, 4.08853466847820432440692271390, 5.16809613374296002304960076102, 5.62109601000354242739842946964, 6.63562521225087766578597788110, 7.58837948844956894517289821597, 8.146627105383223741373878570836
