| L(s) = 1 | − 2.32·2-s − 2.57·4-s − 22.4·7-s + 24.6·8-s − 11·11-s + 9.86·13-s + 52.3·14-s − 36.7·16-s − 128.·17-s + 7.04·19-s + 25.6·22-s + 0.654·23-s − 22.9·26-s + 57.8·28-s + 229.·29-s + 155.·31-s − 111.·32-s + 298.·34-s + 110.·37-s − 16.3·38-s − 154.·41-s + 401.·43-s + 28.3·44-s − 1.52·46-s − 277.·47-s + 161.·49-s − 25.3·52-s + ⋯ |
| L(s) = 1 | − 0.823·2-s − 0.321·4-s − 1.21·7-s + 1.08·8-s − 0.301·11-s + 0.210·13-s + 0.998·14-s − 0.574·16-s − 1.82·17-s + 0.0850·19-s + 0.248·22-s + 0.00593·23-s − 0.173·26-s + 0.390·28-s + 1.46·29-s + 0.902·31-s − 0.615·32-s + 1.50·34-s + 0.489·37-s − 0.0699·38-s − 0.589·41-s + 1.42·43-s + 0.0970·44-s − 0.00488·46-s − 0.861·47-s + 0.471·49-s − 0.0677·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 2.32T + 8T^{2} \) |
| 7 | \( 1 + 22.4T + 343T^{2} \) |
| 13 | \( 1 - 9.86T + 2.19e3T^{2} \) |
| 17 | \( 1 + 128.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 7.04T + 6.85e3T^{2} \) |
| 23 | \( 1 - 0.654T + 1.21e4T^{2} \) |
| 29 | \( 1 - 229.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 155.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 110.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 154.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 401.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 277.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 651.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 423.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 681.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 374.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 96.6T + 3.57e5T^{2} \) |
| 73 | \( 1 - 19.9T + 3.89e5T^{2} \) |
| 79 | \( 1 - 24.4T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.12e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 639.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 730.T + 9.12e5T^{2} \) |
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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
−8.493310181375124655846965060554, −7.54310339024478392100580455064, −6.68287295455919677844783055464, −6.16646293377633518509964780034, −4.87914871996210602206837948778, −4.26488221513102203152465989462, −3.16895847201154798074544776221, −2.19846173862322036140236658924, −0.854566411509687942862874167235, 0,
0.854566411509687942862874167235, 2.19846173862322036140236658924, 3.16895847201154798074544776221, 4.26488221513102203152465989462, 4.87914871996210602206837948778, 6.16646293377633518509964780034, 6.68287295455919677844783055464, 7.54310339024478392100580455064, 8.493310181375124655846965060554
