| L(s) = 1 | + 2.39·2-s + 0.828·3-s + 3.73·4-s + 1.98·6-s + 3.81·7-s + 4.15·8-s − 2.31·9-s − 0.175·11-s + 3.09·12-s + 3.64·13-s + 9.12·14-s + 2.47·16-s + 4.10·17-s − 5.54·18-s − 3.33·19-s + 3.15·21-s − 0.419·22-s + 3.44·23-s + 3.44·24-s + 8.73·26-s − 4.40·27-s + 14.2·28-s − 8.69·29-s + 7.44·31-s − 2.37·32-s − 0.145·33-s + 9.83·34-s + ⋯ |
| L(s) = 1 | + 1.69·2-s + 0.478·3-s + 1.86·4-s + 0.809·6-s + 1.44·7-s + 1.46·8-s − 0.771·9-s − 0.0528·11-s + 0.892·12-s + 1.01·13-s + 2.43·14-s + 0.619·16-s + 0.995·17-s − 1.30·18-s − 0.765·19-s + 0.688·21-s − 0.0894·22-s + 0.718·23-s + 0.702·24-s + 1.71·26-s − 0.847·27-s + 2.69·28-s − 1.61·29-s + 1.33·31-s − 0.419·32-s − 0.0252·33-s + 1.68·34-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
| \(L(1)\) |
\(\approx\) |
\(8.082617533\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.082617533\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.39T + 2T^{2} \) |
| 3 | \( 1 - 0.828T + 3T^{2} \) |
| 7 | \( 1 - 3.81T + 7T^{2} \) |
| 11 | \( 1 + 0.175T + 11T^{2} \) |
| 13 | \( 1 - 3.64T + 13T^{2} \) |
| 17 | \( 1 - 4.10T + 17T^{2} \) |
| 19 | \( 1 + 3.33T + 19T^{2} \) |
| 23 | \( 1 - 3.44T + 23T^{2} \) |
| 29 | \( 1 + 8.69T + 29T^{2} \) |
| 31 | \( 1 - 7.44T + 31T^{2} \) |
| 37 | \( 1 + 4.45T + 37T^{2} \) |
| 41 | \( 1 - 3.61T + 41T^{2} \) |
| 43 | \( 1 - 8.99T + 43T^{2} \) |
| 47 | \( 1 - 13.5T + 47T^{2} \) |
| 53 | \( 1 - 6.28T + 53T^{2} \) |
| 59 | \( 1 - 0.276T + 59T^{2} \) |
| 61 | \( 1 - 8.13T + 61T^{2} \) |
| 67 | \( 1 + 8.85T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 - 8.63T + 73T^{2} \) |
| 79 | \( 1 + 4.87T + 79T^{2} \) |
| 83 | \( 1 - 7.62T + 83T^{2} \) |
| 89 | \( 1 + 4.38T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−7.899506910043678268255801424216, −7.35226735941010282795349880162, −6.37017504409653332274258285041, −5.49066305193566854212313372135, −5.43522517606988141542796580636, −4.24045249737569243407047925209, −3.90269159024443891817039653784, −2.90806142630590368224105573636, −2.26666908500500478059816680114, −1.25633128565702013933820147841,
1.25633128565702013933820147841, 2.26666908500500478059816680114, 2.90806142630590368224105573636, 3.90269159024443891817039653784, 4.24045249737569243407047925209, 5.43522517606988141542796580636, 5.49066305193566854212313372135, 6.37017504409653332274258285041, 7.35226735941010282795349880162, 7.899506910043678268255801424216
