| L(s) = 1 | + 5·2-s + 3·3-s + 17·4-s − 14·5-s + 15·6-s + 32·7-s + 45·8-s + 9·9-s − 70·10-s + 51·12-s + 38·13-s + 160·14-s − 42·15-s + 89·16-s + 2·17-s + 45·18-s − 72·19-s − 238·20-s + 96·21-s + 68·23-s + 135·24-s + 71·25-s + 190·26-s + 27·27-s + 544·28-s + 54·29-s − 210·30-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 0.577·3-s + 17/8·4-s − 1.25·5-s + 1.02·6-s + 1.72·7-s + 1.98·8-s + 1/3·9-s − 2.21·10-s + 1.22·12-s + 0.810·13-s + 3.05·14-s − 0.722·15-s + 1.39·16-s + 0.0285·17-s + 0.589·18-s − 0.869·19-s − 2.66·20-s + 0.997·21-s + 0.616·23-s + 1.14·24-s + 0.567·25-s + 1.43·26-s + 0.192·27-s + 3.67·28-s + 0.345·29-s − 1.27·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.452825845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.452825845\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 5 T + p^{3} T^{2} \) |
| 5 | \( 1 + 14 T + p^{3} T^{2} \) |
| 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
−11.19006251218915211552277133289, −10.85660797934741720806117522488, −8.772090369735353356673906875150, −7.926372491329169539228361320705, −7.21315452598863345440018093497, −5.86733141680236281432592538378, −4.59284218701746206559809718511, −4.20318469982873972522876409409, −3.05566527665222354951417324263, −1.64665048604234698926158528203,
1.64665048604234698926158528203, 3.05566527665222354951417324263, 4.20318469982873972522876409409, 4.59284218701746206559809718511, 5.86733141680236281432592538378, 7.21315452598863345440018093497, 7.926372491329169539228361320705, 8.772090369735353356673906875150, 10.85660797934741720806117522488, 11.19006251218915211552277133289
