L(s) = 1 | − 2.78·2-s − 0.253·4-s + 19.2·5-s + 27.5·7-s + 22.9·8-s − 53.5·10-s + 32.2·11-s + 10.3·13-s − 76.6·14-s − 61.9·16-s + 38.8·17-s + 137.·19-s − 4.88·20-s − 89.6·22-s − 158.·23-s + 245.·25-s − 28.7·26-s − 6.99·28-s − 30.8·29-s + 35.0·31-s − 11.4·32-s − 108.·34-s + 530.·35-s + 105.·37-s − 383.·38-s + 441.·40-s + 295.·41-s + ⋯ |
L(s) = 1 | − 0.984·2-s − 0.0317·4-s + 1.72·5-s + 1.48·7-s + 1.01·8-s − 1.69·10-s + 0.883·11-s + 0.220·13-s − 1.46·14-s − 0.967·16-s + 0.554·17-s + 1.66·19-s − 0.0545·20-s − 0.869·22-s − 1.43·23-s + 1.96·25-s − 0.217·26-s − 0.0472·28-s − 0.197·29-s + 0.202·31-s − 0.0634·32-s − 0.545·34-s + 2.55·35-s + 0.466·37-s − 1.63·38-s + 1.74·40-s + 1.12·41-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
\(L(2)\) |
\(\approx\) |
\(2.717067390\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.717067390\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + 2.78T + 8T^{2} \) |
| 5 | \( 1 - 19.2T + 125T^{2} \) |
| 7 | \( 1 - 27.5T + 343T^{2} \) |
| 11 | \( 1 - 32.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 10.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 38.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 137.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 158.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 30.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 35.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 105.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 295.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 311.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 122.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 700.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 548.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 206.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 352.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 759.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 587.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 149.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.02e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.04e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.80e3T + 9.12e5T^{2} \) |
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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
−8.862564346557258184828207353775, −8.077028462148470304902250970870, −7.46277242814013632822479431395, −6.39213418407678085362818827772, −5.51334803093062347370453051481, −4.95459727861591273239705824342, −3.88025302749753763263594366174, −2.32108452608543222194322801369, −1.48491325846575055526299433536, −1.03797391223964977991131008989,
1.03797391223964977991131008989, 1.48491325846575055526299433536, 2.32108452608543222194322801369, 3.88025302749753763263594366174, 4.95459727861591273239705824342, 5.51334803093062347370453051481, 6.39213418407678085362818827772, 7.46277242814013632822479431395, 8.077028462148470304902250970870, 8.862564346557258184828207353775