| L(s) = 1 | + 1.80·2-s + 2.24·4-s − 0.149·5-s + 2.24·8-s − 0.269·10-s + 11-s + 1.80·16-s − 1.46·19-s − 0.335·20-s + 1.80·22-s + 1.97·23-s − 0.977·25-s − 0.730·29-s + 1.00·32-s − 2.64·38-s − 0.335·40-s − 1.24·41-s + 2.24·44-s + 3.56·46-s + 49-s − 1.76·50-s + 1.46·53-s − 0.149·55-s − 1.31·58-s − 1.65·59-s − 0.445·61-s − 3.29·76-s + ⋯ |
| L(s) = 1 | + 1.80·2-s + 2.24·4-s − 0.149·5-s + 2.24·8-s − 0.269·10-s + 11-s + 1.80·16-s − 1.46·19-s − 0.335·20-s + 1.80·22-s + 1.97·23-s − 0.977·25-s − 0.730·29-s + 1.00·32-s − 2.64·38-s − 0.335·40-s − 1.24·41-s + 2.24·44-s + 3.56·46-s + 49-s − 1.76·50-s + 1.46·53-s − 0.149·55-s − 1.31·58-s − 1.65·59-s − 0.445·61-s − 3.29·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3879 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3879 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.876382779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.876382779\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 431 | \( 1 + T \) |
| good | 2 | \( 1 - 1.80T + T^{2} \) |
| 5 | \( 1 + 0.149T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 1.46T + T^{2} \) |
| 23 | \( 1 - 1.97T + T^{2} \) |
| 29 | \( 1 + 0.730T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.24T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.46T + T^{2} \) |
| 59 | \( 1 + 1.65T + T^{2} \) |
| 61 | \( 1 + 0.445T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - 0.730T + T^{2} \) |
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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
−8.641025028108646912998157514318, −7.53341143780723260438206023992, −6.87336625958448685319475524383, −6.32273649796976267232479463703, −5.57042077522227248172366571935, −4.78495840846498586340328154327, −4.08068733553433030359700021590, −3.49765066622665831777295518464, −2.54010379874287977800098460668, −1.57978292464358795259905939985,
1.57978292464358795259905939985, 2.54010379874287977800098460668, 3.49765066622665831777295518464, 4.08068733553433030359700021590, 4.78495840846498586340328154327, 5.57042077522227248172366571935, 6.32273649796976267232479463703, 6.87336625958448685319475524383, 7.53341143780723260438206023992, 8.641025028108646912998157514318
