| L(s) = 1 | + 0.641·2-s − 1.91·3-s − 7.58·4-s + 0.642·5-s − 1.22·6-s + 23.2·7-s − 9.99·8-s − 23.3·9-s + 0.411·10-s − 11·11-s + 14.5·12-s + 14.8·14-s − 1.22·15-s + 54.2·16-s − 35.1·17-s − 14.9·18-s + 37.3·19-s − 4.87·20-s − 44.4·21-s − 7.05·22-s + 121.·23-s + 19.1·24-s − 124.·25-s + 96.3·27-s − 176.·28-s − 19.7·29-s − 0.788·30-s + ⋯ |
| L(s) = 1 | + 0.226·2-s − 0.368·3-s − 0.948·4-s + 0.0574·5-s − 0.0834·6-s + 1.25·7-s − 0.441·8-s − 0.864·9-s + 0.0130·10-s − 0.301·11-s + 0.349·12-s + 0.284·14-s − 0.0211·15-s + 0.848·16-s − 0.502·17-s − 0.195·18-s + 0.451·19-s − 0.0545·20-s − 0.461·21-s − 0.0683·22-s + 1.10·23-s + 0.162·24-s − 0.996·25-s + 0.686·27-s − 1.19·28-s − 0.126·29-s − 0.00479·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.641T + 8T^{2} \) |
| 3 | \( 1 + 1.91T + 27T^{2} \) |
| 5 | \( 1 - 0.642T + 125T^{2} \) |
| 7 | \( 1 - 23.2T + 343T^{2} \) |
| 17 | \( 1 + 35.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 37.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 121.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 19.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 30.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 193.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 80.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 196.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 182.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 451.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 270.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 694.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 364.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 772.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 160.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 46.4T + 4.93e5T^{2} \) |
| 83 | \( 1 + 520.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 394.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 17.2T + 9.12e5T^{2} \) |
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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
−8.500076957725515168410184027275, −7.911604923287478156547847355584, −6.88908071039659223806313683153, −5.68291329671982061126703647236, −5.26582578626351439165066833610, −4.55887661666210738648536287701, −3.58288330334626811770985267191, −2.43123965403167903941082102509, −1.12046292717019166173503483044, 0,
1.12046292717019166173503483044, 2.43123965403167903941082102509, 3.58288330334626811770985267191, 4.55887661666210738648536287701, 5.26582578626351439165066833610, 5.68291329671982061126703647236, 6.88908071039659223806313683153, 7.911604923287478156547847355584, 8.500076957725515168410184027275
