| L(s) = 1 | − 3·3-s + 5·5-s + 13.7·7-s + 9·9-s − 12.6·11-s + 91.3·13-s − 15·15-s − 135.·17-s − 117.·19-s − 41.2·21-s + 23·23-s + 25·25-s − 27·27-s − 243.·29-s + 123.·31-s + 38.0·33-s + 68.7·35-s + 36.5·37-s − 273.·39-s − 356.·41-s + 232.·43-s + 45·45-s − 548.·47-s − 153.·49-s + 406.·51-s + 647.·53-s − 63.4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.742·7-s + 0.333·9-s − 0.347·11-s + 1.94·13-s − 0.258·15-s − 1.93·17-s − 1.41·19-s − 0.428·21-s + 0.208·23-s + 0.200·25-s − 0.192·27-s − 1.55·29-s + 0.716·31-s + 0.200·33-s + 0.332·35-s + 0.162·37-s − 1.12·39-s − 1.35·41-s + 0.824·43-s + 0.149·45-s − 1.70·47-s − 0.448·49-s + 1.11·51-s + 1.67·53-s − 0.155·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 - 23T \) |
| good | 7 | \( 1 - 13.7T + 343T^{2} \) |
| 11 | \( 1 + 12.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 91.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 135.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 117.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 243.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 123.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 36.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 356.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 232.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 548.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 647.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 125.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 131.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 432.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 429.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 388.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.19e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 846.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.37e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.02e3T + 9.12e5T^{2} \) |
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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
−8.590267465255745593484812156181, −8.313475252687655625537467183753, −6.89835458618153651967995085966, −6.34765462620874209599923090999, −5.53249601973221738984790169316, −4.57967863537527359718209584148, −3.81105449701782532882532183640, −2.25936347757432756698028100261, −1.42146854031802900942191609433, 0,
1.42146854031802900942191609433, 2.25936347757432756698028100261, 3.81105449701782532882532183640, 4.57967863537527359718209584148, 5.53249601973221738984790169316, 6.34765462620874209599923090999, 6.89835458618153651967995085966, 8.313475252687655625537467183753, 8.590267465255745593484812156181
