| L(s) = 1 | − 2.35·2-s − 2.43·4-s − 3.14·5-s − 17.8·7-s + 24.6·8-s + 7.42·10-s − 0.0693·11-s + 41.9·13-s + 42.0·14-s − 38.6·16-s − 23.5·17-s + 90.8·19-s + 7.66·20-s + 0.163·22-s − 139.·23-s − 115.·25-s − 98.9·26-s + 43.3·28-s − 86.6·29-s + 13.4·31-s − 105.·32-s + 55.6·34-s + 56.0·35-s + 199.·37-s − 214.·38-s − 77.4·40-s − 229.·41-s + ⋯ |
| L(s) = 1 | − 0.834·2-s − 0.304·4-s − 0.281·5-s − 0.961·7-s + 1.08·8-s + 0.234·10-s − 0.00190·11-s + 0.894·13-s + 0.801·14-s − 0.603·16-s − 0.336·17-s + 1.09·19-s + 0.0856·20-s + 0.00158·22-s − 1.26·23-s − 0.920·25-s − 0.746·26-s + 0.292·28-s − 0.554·29-s + 0.0778·31-s − 0.584·32-s + 0.280·34-s + 0.270·35-s + 0.886·37-s − 0.914·38-s − 0.306·40-s − 0.875·41-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 | \( 1 + 2.35T + 8T^{2} \) |
| 5 | \( 1 + 3.14T + 125T^{2} \) |
| 7 | \( 1 + 17.8T + 343T^{2} \) |
| 11 | \( 1 + 0.0693T + 1.33e3T^{2} \) |
| 13 | \( 1 - 41.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 23.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 90.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 139.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 86.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 13.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 199.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 229.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 273.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 419.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 250.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 514.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 754.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 224.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 343.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 702.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 113.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.07e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 898.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 424.T + 9.12e5T^{2} \) |
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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
−8.294678561119645952498567242513, −7.84295190910585258261325359263, −6.91676002042848158502380563434, −6.09431529721160893395209020655, −5.20519928772254356026171061143, −4.01824181723397287899794630382, −3.52325207425720345606341812091, −2.12469141516802787088143870024, −0.928830691477130055138630178256, 0,
0.928830691477130055138630178256, 2.12469141516802787088143870024, 3.52325207425720345606341812091, 4.01824181723397287899794630382, 5.20519928772254356026171061143, 6.09431529721160893395209020655, 6.91676002042848158502380563434, 7.84295190910585258261325359263, 8.294678561119645952498567242513
