| L(s) = 1 | + 2.31·3-s + 6.10·5-s − 4.57·7-s − 21.6·9-s + 47.5·13-s + 14.1·15-s − 63.6·17-s − 18.4·19-s − 10.5·21-s − 51.9·23-s − 87.7·25-s − 112.·27-s + 77.1·29-s + 109.·31-s − 27.9·35-s − 139.·37-s + 110.·39-s + 158.·41-s + 26.9·43-s − 131.·45-s − 258.·47-s − 322.·49-s − 147.·51-s − 148.·53-s − 42.7·57-s − 725.·59-s − 482.·61-s + ⋯ |
| L(s) = 1 | + 0.446·3-s + 0.545·5-s − 0.246·7-s − 0.801·9-s + 1.01·13-s + 0.243·15-s − 0.907·17-s − 0.222·19-s − 0.110·21-s − 0.470·23-s − 0.702·25-s − 0.803·27-s + 0.494·29-s + 0.632·31-s − 0.134·35-s − 0.621·37-s + 0.452·39-s + 0.603·41-s + 0.0954·43-s − 0.437·45-s − 0.801·47-s − 0.939·49-s − 0.404·51-s − 0.383·53-s − 0.0993·57-s − 1.60·59-s − 1.01·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2.31T + 27T^{2} \) |
| 5 | \( 1 - 6.10T + 125T^{2} \) |
| 7 | \( 1 + 4.57T + 343T^{2} \) |
| 13 | \( 1 - 47.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 63.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 18.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 51.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 77.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 109.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 139.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 158.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 26.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 258.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 148.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 725.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 482.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 464.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 912.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 917.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 759.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 698.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 564.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.16e3T + 9.12e5T^{2} \) |
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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
−9.117594225424453294753263717043, −8.511727714492069974788728344829, −7.70601778562587214935608358192, −6.36753396874683770259723914641, −6.02665647192699874197266937284, −4.77886845999885543229021135351, −3.65365201529648490327149055823, −2.69199020785660166340275213416, −1.63498312285657234579690254152, 0,
1.63498312285657234579690254152, 2.69199020785660166340275213416, 3.65365201529648490327149055823, 4.77886845999885543229021135351, 6.02665647192699874197266937284, 6.36753396874683770259723914641, 7.70601778562587214935608358192, 8.511727714492069974788728344829, 9.117594225424453294753263717043
