| L(s) = 1 | − 20.3·5-s + 7·7-s + 30.9·11-s + 50.6·13-s + 102.·17-s − 61.2·19-s − 148.·23-s + 287.·25-s − 159.·29-s − 121.·31-s − 142.·35-s − 357.·37-s − 466.·41-s − 185.·43-s + 131.·47-s + 49·49-s − 200.·53-s − 627.·55-s + 591.·59-s + 70.5·61-s − 1.02e3·65-s − 643.·67-s + 522.·71-s − 576.·73-s + 216.·77-s + 280.·79-s + 557.·83-s + ⋯ |
| L(s) = 1 | − 1.81·5-s + 0.377·7-s + 0.847·11-s + 1.07·13-s + 1.46·17-s − 0.739·19-s − 1.34·23-s + 2.29·25-s − 1.01·29-s − 0.702·31-s − 0.686·35-s − 1.59·37-s − 1.77·41-s − 0.658·43-s + 0.407·47-s + 0.142·49-s − 0.518·53-s − 1.53·55-s + 1.30·59-s + 0.148·61-s − 1.96·65-s − 1.17·67-s + 0.873·71-s − 0.923·73-s + 0.320·77-s + 0.399·79-s + 0.737·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 5 | \( 1 + 20.3T + 125T^{2} \) |
| 11 | \( 1 - 30.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 50.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 102.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 61.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 148.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 159.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 121.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 357.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 466.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 185.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 131.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 200.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 591.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 70.5T + 2.26e5T^{2} \) |
| 67 | \( 1 + 643.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 522.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 576.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 280.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 557.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.22e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 65.0T + 9.12e5T^{2} \) |
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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
−10.25766199974540252842845590230, −8.872034337651921528095099355461, −8.227884692787713696896016997207, −7.50628729151117848897611452819, −6.50140004726001976394573395993, −5.21944881798502008165559737448, −3.83930087190104098589088187740, −3.62111958497446769258211289943, −1.48465324693089921411677507212, 0,
1.48465324693089921411677507212, 3.62111958497446769258211289943, 3.83930087190104098589088187740, 5.21944881798502008165559737448, 6.50140004726001976394573395993, 7.50628729151117848897611452819, 8.227884692787713696896016997207, 8.872034337651921528095099355461, 10.25766199974540252842845590230
