| L(s) = 1 | + 1.05·2-s − 4.53·3-s − 6.88·4-s − 14.4·5-s − 4.78·6-s − 15.4·7-s − 15.6·8-s − 6.42·9-s − 15.2·10-s − 62.4·11-s + 31.2·12-s − 21.6·13-s − 16.3·14-s + 65.6·15-s + 38.5·16-s + 25.1·17-s − 6.77·18-s + 45.2·19-s + 99.6·20-s + 70.1·21-s − 65.8·22-s + 195.·23-s + 71.2·24-s + 84.4·25-s − 22.7·26-s + 151.·27-s + 106.·28-s + ⋯ |
| L(s) = 1 | + 0.372·2-s − 0.872·3-s − 0.861·4-s − 1.29·5-s − 0.325·6-s − 0.835·7-s − 0.693·8-s − 0.237·9-s − 0.482·10-s − 1.71·11-s + 0.751·12-s − 0.461·13-s − 0.311·14-s + 1.13·15-s + 0.602·16-s + 0.358·17-s − 0.0886·18-s + 0.546·19-s + 1.11·20-s + 0.729·21-s − 0.638·22-s + 1.76·23-s + 0.605·24-s + 0.675·25-s − 0.171·26-s + 1.08·27-s + 0.719·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
| good | 2 | \( 1 - 1.05T + 8T^{2} \) |
| 3 | \( 1 + 4.53T + 27T^{2} \) |
| 5 | \( 1 + 14.4T + 125T^{2} \) |
| 7 | \( 1 + 15.4T + 343T^{2} \) |
| 11 | \( 1 + 62.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 21.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 25.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 45.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 195.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 39.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 254.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 110.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 62.1T + 6.89e4T^{2} \) |
| 47 | \( 1 - 586.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 515.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 250.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 812.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 558.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 227.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 568.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.33e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 16.2T + 5.71e5T^{2} \) |
| 89 | \( 1 - 858.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 312.T + 9.12e5T^{2} \) |
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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
−8.474783620498820384799845049241, −7.60021452049986626649225948709, −7.00349411400900112997897044229, −5.67701264576644834715819454933, −5.29942100283029719334283825407, −4.53113425308273676054294331914, −3.42356732966095042715548034622, −2.89531892146267553482947843292, −0.61406931325781491862928900344, 0,
0.61406931325781491862928900344, 2.89531892146267553482947843292, 3.42356732966095042715548034622, 4.53113425308273676054294331914, 5.29942100283029719334283825407, 5.67701264576644834715819454933, 7.00349411400900112997897044229, 7.60021452049986626649225948709, 8.474783620498820384799845049241
