| L(s) = 1 | + 0.0765·2-s − 3·3-s − 7.99·4-s − 0.229·6-s − 7·7-s − 1.22·8-s + 9·9-s + 10.8·11-s + 23.9·12-s + 26.5·13-s − 0.535·14-s + 63.8·16-s + 95.1·17-s + 0.688·18-s − 35.4·19-s + 21·21-s + 0.833·22-s − 62.8·23-s + 3.67·24-s + 2.03·26-s − 27·27-s + 55.9·28-s + 117.·29-s − 171.·31-s + 14.6·32-s − 32.6·33-s + 7.27·34-s + ⋯ |
| L(s) = 1 | + 0.0270·2-s − 0.577·3-s − 0.999·4-s − 0.0156·6-s − 0.377·7-s − 0.0540·8-s + 0.333·9-s + 0.298·11-s + 0.576·12-s + 0.566·13-s − 0.0102·14-s + 0.997·16-s + 1.35·17-s + 0.00901·18-s − 0.428·19-s + 0.218·21-s + 0.00807·22-s − 0.569·23-s + 0.0312·24-s + 0.0153·26-s − 0.192·27-s + 0.377·28-s + 0.754·29-s − 0.991·31-s + 0.0810·32-s − 0.172·33-s + 0.0367·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 0.0765T + 8T^{2} \) |
| 11 | \( 1 - 10.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 26.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 95.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 35.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 62.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 117.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 171.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 203.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 428.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 96.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 407.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 380.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 287.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 823.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 585.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 653.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.05e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 751.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 844.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 262.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 814.T + 9.12e5T^{2} \) |
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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
−10.08571012750560074252971917483, −9.153189119637232217024849436063, −8.344422811151576644505828919399, −7.26876744289457349864953095758, −6.08352047131292792363763728400, −5.37238307741542573313414234779, −4.23778054719692508183117261549, −3.33780095178602100439616431295, −1.32514264985454957977737256004, 0,
1.32514264985454957977737256004, 3.33780095178602100439616431295, 4.23778054719692508183117261549, 5.37238307741542573313414234779, 6.08352047131292792363763728400, 7.26876744289457349864953095758, 8.344422811151576644505828919399, 9.153189119637232217024849436063, 10.08571012750560074252971917483
