| L(s) = 1 | + 1.23·2-s − 0.485·4-s + 3.43·5-s + 5.19·7-s − 3.05·8-s + 4.23·10-s + 1.70·11-s − 2.08·13-s + 6.39·14-s − 2.79·16-s + 4.47·17-s + 19-s − 1.66·20-s + 2.10·22-s + 5.73·23-s + 6.83·25-s − 2.56·26-s − 2.52·28-s − 4.18·29-s − 4.77·31-s + 2.67·32-s + 5.50·34-s + 17.8·35-s + 3.37·37-s + 1.23·38-s − 10.5·40-s − 5.54·41-s + ⋯ |
| L(s) = 1 | + 0.870·2-s − 0.242·4-s + 1.53·5-s + 1.96·7-s − 1.08·8-s + 1.33·10-s + 0.514·11-s − 0.577·13-s + 1.71·14-s − 0.698·16-s + 1.08·17-s + 0.229·19-s − 0.373·20-s + 0.447·22-s + 1.19·23-s + 1.36·25-s − 0.502·26-s − 0.476·28-s − 0.777·29-s − 0.857·31-s + 0.473·32-s + 0.944·34-s + 3.02·35-s + 0.554·37-s + 0.199·38-s − 1.66·40-s − 0.865·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.266410547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.266410547\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - 1.23T + 2T^{2} \) |
| 5 | \( 1 - 3.43T + 5T^{2} \) |
| 7 | \( 1 - 5.19T + 7T^{2} \) |
| 11 | \( 1 - 1.70T + 11T^{2} \) |
| 13 | \( 1 + 2.08T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 23 | \( 1 - 5.73T + 23T^{2} \) |
| 29 | \( 1 + 4.18T + 29T^{2} \) |
| 31 | \( 1 + 4.77T + 31T^{2} \) |
| 37 | \( 1 - 3.37T + 37T^{2} \) |
| 41 | \( 1 + 5.54T + 41T^{2} \) |
| 43 | \( 1 - 7.86T + 43T^{2} \) |
| 53 | \( 1 + 6.19T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 - 9.61T + 61T^{2} \) |
| 67 | \( 1 + 2.89T + 67T^{2} \) |
| 71 | \( 1 - 2.66T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + 8.63T + 83T^{2} \) |
| 89 | \( 1 + 1.55T + 89T^{2} \) |
| 97 | \( 1 - 15.3T + 97T^{2} \) |
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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
−7.67426331914238085626630286439, −7.16828255568089188540198228396, −5.94344073175695186718098003030, −5.71114194255486534906125968167, −4.92144923883698036463038191809, −4.68110867365709304932148718983, −3.60559788355398191874079786851, −2.66692325183240712917145632357, −1.81109493792923213717582949445, −1.11464580962640751662876076056,
1.11464580962640751662876076056, 1.81109493792923213717582949445, 2.66692325183240712917145632357, 3.60559788355398191874079786851, 4.68110867365709304932148718983, 4.92144923883698036463038191809, 5.71114194255486534906125968167, 5.94344073175695186718098003030, 7.16828255568089188540198228396, 7.67426331914238085626630286439
