| L(s) = 1 | − 2.40·2-s − 2.30·3-s + 3.78·4-s + 5.54·6-s + 4.72·7-s − 4.29·8-s + 2.31·9-s + 1.32·11-s − 8.73·12-s − 2.94·13-s − 11.3·14-s + 2.76·16-s − 2.94·17-s − 5.57·18-s + 5.65·19-s − 10.9·21-s − 3.19·22-s − 5.25·23-s + 9.91·24-s + 7.08·26-s + 1.57·27-s + 17.9·28-s + 6.80·29-s + 4.16·31-s + 1.94·32-s − 3.06·33-s + 7.07·34-s + ⋯ |
| L(s) = 1 | − 1.70·2-s − 1.33·3-s + 1.89·4-s + 2.26·6-s + 1.78·7-s − 1.51·8-s + 0.773·9-s + 0.400·11-s − 2.52·12-s − 0.817·13-s − 3.03·14-s + 0.691·16-s − 0.713·17-s − 1.31·18-s + 1.29·19-s − 2.37·21-s − 0.681·22-s − 1.09·23-s + 2.02·24-s + 1.39·26-s + 0.302·27-s + 3.38·28-s + 1.26·29-s + 0.747·31-s + 0.342·32-s − 0.533·33-s + 1.21·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 + 2.40T + 2T^{2} \) |
| 3 | \( 1 + 2.30T + 3T^{2} \) |
| 7 | \( 1 - 4.72T + 7T^{2} \) |
| 11 | \( 1 - 1.32T + 11T^{2} \) |
| 13 | \( 1 + 2.94T + 13T^{2} \) |
| 17 | \( 1 + 2.94T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 + 5.25T + 23T^{2} \) |
| 29 | \( 1 - 6.80T + 29T^{2} \) |
| 31 | \( 1 - 4.16T + 31T^{2} \) |
| 37 | \( 1 - 1.74T + 37T^{2} \) |
| 41 | \( 1 + 7.98T + 41T^{2} \) |
| 43 | \( 1 - 0.0704T + 43T^{2} \) |
| 47 | \( 1 + 3.91T + 47T^{2} \) |
| 53 | \( 1 + 8.86T + 53T^{2} \) |
| 59 | \( 1 + 4.11T + 59T^{2} \) |
| 61 | \( 1 + 1.61T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 9.46T + 71T^{2} \) |
| 73 | \( 1 + 3.02T + 73T^{2} \) |
| 79 | \( 1 + 7.08T + 79T^{2} \) |
| 83 | \( 1 - 1.44T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 + 6.93T + 97T^{2} \) |
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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
−7.971080020334441633351038098103, −7.11539660733462868044801564426, −6.57015225878082069412142064243, −5.72414812456712977353227402812, −4.88156262214733569100925362852, −4.45647303726556972966775623670, −2.75682926894984959763878820750, −1.70913689871395676706880705130, −1.11632102165334255523636829347, 0,
1.11632102165334255523636829347, 1.70913689871395676706880705130, 2.75682926894984959763878820750, 4.45647303726556972966775623670, 4.88156262214733569100925362852, 5.72414812456712977353227402812, 6.57015225878082069412142064243, 7.11539660733462868044801564426, 7.971080020334441633351038098103
