| L(s) = 1 | + 1.38·2-s − 0.0912·4-s + 1.35·5-s − 1.50·7-s − 2.88·8-s + 1.86·10-s + 5.20·11-s − 6.03·13-s − 2.07·14-s − 3.80·16-s − 2.23·17-s − 5.35·19-s − 0.123·20-s + 7.19·22-s − 2.95·23-s − 3.17·25-s − 8.34·26-s + 0.137·28-s + 0.529·29-s − 3.72·31-s + 0.515·32-s − 3.08·34-s − 2.03·35-s + 5.79·37-s − 7.40·38-s − 3.90·40-s + 4.00·41-s + ⋯ |
| L(s) = 1 | + 0.976·2-s − 0.0456·4-s + 0.604·5-s − 0.568·7-s − 1.02·8-s + 0.590·10-s + 1.57·11-s − 1.67·13-s − 0.555·14-s − 0.952·16-s − 0.542·17-s − 1.22·19-s − 0.0275·20-s + 1.53·22-s − 0.616·23-s − 0.634·25-s − 1.63·26-s + 0.0259·28-s + 0.0982·29-s − 0.668·31-s + 0.0911·32-s − 0.529·34-s − 0.343·35-s + 0.952·37-s − 1.20·38-s − 0.617·40-s + 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + T \) |
| good | 2 | \( 1 - 1.38T + 2T^{2} \) |
| 5 | \( 1 - 1.35T + 5T^{2} \) |
| 7 | \( 1 + 1.50T + 7T^{2} \) |
| 11 | \( 1 - 5.20T + 11T^{2} \) |
| 13 | \( 1 + 6.03T + 13T^{2} \) |
| 17 | \( 1 + 2.23T + 17T^{2} \) |
| 19 | \( 1 + 5.35T + 19T^{2} \) |
| 23 | \( 1 + 2.95T + 23T^{2} \) |
| 29 | \( 1 - 0.529T + 29T^{2} \) |
| 31 | \( 1 + 3.72T + 31T^{2} \) |
| 37 | \( 1 - 5.79T + 37T^{2} \) |
| 41 | \( 1 - 4.00T + 41T^{2} \) |
| 43 | \( 1 - 4.50T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 - 8.22T + 53T^{2} \) |
| 59 | \( 1 + 5.35T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 + 9.35T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 5.72T + 79T^{2} \) |
| 83 | \( 1 - 4.70T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−8.980268442854726701948695581515, −7.84433607754896367213430927799, −6.67202764679825872444788152746, −6.32543110919450094616692261084, −5.48880752230952095578416701523, −4.46324747300008663848214555266, −4.02115766674652948604696470076, −2.86706043048514136773633372796, −1.93739781660191230361709662110, 0,
1.93739781660191230361709662110, 2.86706043048514136773633372796, 4.02115766674652948604696470076, 4.46324747300008663848214555266, 5.48880752230952095578416701523, 6.32543110919450094616692261084, 6.67202764679825872444788152746, 7.84433607754896367213430927799, 8.980268442854726701948695581515
