| L(s) = 1 | + 5·2-s − 3·3-s + 17·4-s − 15·6-s + 32·7-s + 45·8-s + 9·9-s − 11·11-s − 51·12-s + 38·13-s + 160·14-s + 89·16-s + 2·17-s + 45·18-s + 72·19-s − 96·21-s − 55·22-s − 68·23-s − 135·24-s + 190·26-s − 27·27-s + 544·28-s − 54·29-s − 152·31-s + 85·32-s + 33·33-s + 10·34-s + ⋯ |
| L(s) = 1 | + 1.76·2-s − 0.577·3-s + 17/8·4-s − 1.02·6-s + 1.72·7-s + 1.98·8-s + 1/3·9-s − 0.301·11-s − 1.22·12-s + 0.810·13-s + 3.05·14-s + 1.39·16-s + 0.0285·17-s + 0.589·18-s + 0.869·19-s − 0.997·21-s − 0.533·22-s − 0.616·23-s − 1.14·24-s + 1.43·26-s − 0.192·27-s + 3.67·28-s − 0.345·29-s − 0.880·31-s + 0.469·32-s + 0.174·33-s + 0.0504·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.621813270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.621813270\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + p T \) |
| good | 2 | \( 1 - 5 T + p^{3} T^{2} \) |
| 7 | \( 1 - 32 T + p^{3} T^{2} \) |
| 13 | \( 1 - 38 T + p^{3} T^{2} \) |
| 17 | \( 1 - 2 T + p^{3} T^{2} \) |
| 19 | \( 1 - 72 T + p^{3} T^{2} \) |
| 23 | \( 1 + 68 T + p^{3} T^{2} \) |
| 29 | \( 1 + 54 T + p^{3} T^{2} \) |
| 31 | \( 1 + 152 T + p^{3} T^{2} \) |
| 37 | \( 1 + 174 T + p^{3} T^{2} \) |
| 41 | \( 1 - 94 T + p^{3} T^{2} \) |
| 43 | \( 1 - 528 T + p^{3} T^{2} \) |
| 47 | \( 1 - 340 T + p^{3} T^{2} \) |
| 53 | \( 1 - 438 T + p^{3} T^{2} \) |
| 59 | \( 1 - 20 T + p^{3} T^{2} \) |
| 61 | \( 1 - 570 T + p^{3} T^{2} \) |
| 67 | \( 1 - 460 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1092 T + p^{3} T^{2} \) |
| 73 | \( 1 + 562 T + p^{3} T^{2} \) |
| 79 | \( 1 + 16 T + p^{3} T^{2} \) |
| 83 | \( 1 + 372 T + p^{3} T^{2} \) |
| 89 | \( 1 + 966 T + p^{3} T^{2} \) |
| 97 | \( 1 - 526 T + p^{3} T^{2} \) |
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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
−10.40595143830431694213157730555, −8.849575151895445644053520394121, −7.71893458991469003236997029698, −7.08084679348675688222525032054, −5.69543625182614946043289515543, −5.51895382976444059682466037194, −4.47599014722019778538823975142, −3.80200343943497455956426654618, −2.36130647763412951757279325389, −1.29672241045456255008527671981,
1.29672241045456255008527671981, 2.36130647763412951757279325389, 3.80200343943497455956426654618, 4.47599014722019778538823975142, 5.51895382976444059682466037194, 5.69543625182614946043289515543, 7.08084679348675688222525032054, 7.71893458991469003236997029698, 8.849575151895445644053520394121, 10.40595143830431694213157730555
