| L(s) = 1 | − 5·2-s + 17·4-s − 5·5-s + 7·7-s − 45·8-s + 25·10-s − 12·11-s + 30·13-s − 35·14-s + 89·16-s + 134·17-s − 92·19-s − 85·20-s + 60·22-s − 112·23-s + 25·25-s − 150·26-s + 119·28-s + 58·29-s − 224·31-s − 85·32-s − 670·34-s − 35·35-s − 146·37-s + 460·38-s + 225·40-s − 18·41-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 17/8·4-s − 0.447·5-s + 0.377·7-s − 1.98·8-s + 0.790·10-s − 0.328·11-s + 0.640·13-s − 0.668·14-s + 1.39·16-s + 1.91·17-s − 1.11·19-s − 0.950·20-s + 0.581·22-s − 1.01·23-s + 1/5·25-s − 1.13·26-s + 0.803·28-s + 0.371·29-s − 1.29·31-s − 0.469·32-s − 3.37·34-s − 0.169·35-s − 0.648·37-s + 1.96·38-s + 0.889·40-s − 0.0685·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6951876293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6951876293\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 - p T \) |
| good | 2 | \( 1 + 5 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 30 T + p^{3} T^{2} \) |
| 17 | \( 1 - 134 T + p^{3} T^{2} \) |
| 19 | \( 1 + 92 T + p^{3} T^{2} \) |
| 23 | \( 1 + 112 T + p^{3} T^{2} \) |
| 29 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 31 | \( 1 + 224 T + p^{3} T^{2} \) |
| 37 | \( 1 + 146 T + p^{3} T^{2} \) |
| 41 | \( 1 + 18 T + p^{3} T^{2} \) |
| 43 | \( 1 - 340 T + p^{3} T^{2} \) |
| 47 | \( 1 + 208 T + p^{3} T^{2} \) |
| 53 | \( 1 - 754 T + p^{3} T^{2} \) |
| 59 | \( 1 + 380 T + p^{3} T^{2} \) |
| 61 | \( 1 - 718 T + p^{3} T^{2} \) |
| 67 | \( 1 - 412 T + p^{3} T^{2} \) |
| 71 | \( 1 - 960 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1066 T + p^{3} T^{2} \) |
| 79 | \( 1 - 896 T + p^{3} T^{2} \) |
| 83 | \( 1 + 436 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1038 T + p^{3} T^{2} \) |
| 97 | \( 1 + 702 T + p^{3} T^{2} \) |
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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
−10.85825795867688488784881364427, −10.28422601790192977845644585863, −9.293599220545329796901925178813, −8.256730079759568715571231017889, −7.86242577601528338089183343274, −6.78234663194193976534017188524, −5.55833835918804244684565009021, −3.70409121002516072506981423307, −2.08777255820772856016681214572, −0.74893114859549830706408251612,
0.74893114859549830706408251612, 2.08777255820772856016681214572, 3.70409121002516072506981423307, 5.55833835918804244684565009021, 6.78234663194193976534017188524, 7.86242577601528338089183343274, 8.256730079759568715571231017889, 9.293599220545329796901925178813, 10.28422601790192977845644585863, 10.85825795867688488784881364427
