L(s) = 1 | + 3.24·2-s + 2.55·4-s + 21.6·5-s + 3.86·7-s − 17.6·8-s + 70.3·10-s − 54.0·11-s + 5.53·13-s + 12.5·14-s − 77.9·16-s − 13.4·17-s − 72.3·19-s + 55.4·20-s − 175.·22-s + 53.2·23-s + 343.·25-s + 17.9·26-s + 9.90·28-s − 267.·29-s − 239.·31-s − 111.·32-s − 43.7·34-s + 83.7·35-s − 139.·37-s − 235.·38-s − 382.·40-s + 0.898·41-s + ⋯ |
L(s) = 1 | + 1.14·2-s + 0.319·4-s + 1.93·5-s + 0.208·7-s − 0.781·8-s + 2.22·10-s − 1.48·11-s + 0.118·13-s + 0.239·14-s − 1.21·16-s − 0.192·17-s − 0.873·19-s + 0.619·20-s − 1.70·22-s + 0.482·23-s + 2.74·25-s + 0.135·26-s + 0.0668·28-s − 1.71·29-s − 1.38·31-s − 0.617·32-s − 0.220·34-s + 0.404·35-s − 0.618·37-s − 1.00·38-s − 1.51·40-s + 0.00342·41-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 | \( 1 - 3.24T + 8T^{2} \) |
| 5 | \( 1 - 21.6T + 125T^{2} \) |
| 7 | \( 1 - 3.86T + 343T^{2} \) |
| 11 | \( 1 + 54.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 5.53T + 2.19e3T^{2} \) |
| 17 | \( 1 + 13.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 72.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 53.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 267.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 239.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 139.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 0.898T + 6.89e4T^{2} \) |
| 43 | \( 1 - 366.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 100.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 704.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 808.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 466.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 266.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 358.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 174.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 674.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 656.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 380.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 946.T + 9.12e5T^{2} \) |
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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
−8.483792560815616578133264345475, −7.30447964043128054561882024838, −6.39321406625351993507221243835, −5.64020247815935567834255071982, −5.32884199709195547558187627514, −4.52742369916759320475590903845, −3.29249155010655570058776246056, −2.42710556044022639749357384224, −1.75194533297549395309699315277, 0,
1.75194533297549395309699315277, 2.42710556044022639749357384224, 3.29249155010655570058776246056, 4.52742369916759320475590903845, 5.32884199709195547558187627514, 5.64020247815935567834255071982, 6.39321406625351993507221243835, 7.30447964043128054561882024838, 8.483792560815616578133264345475