| L(s) = 1 | − 3.55·2-s − 3.90·3-s + 4.67·4-s − 19.3·5-s + 13.8·6-s + 18.4·7-s + 11.8·8-s − 11.7·9-s + 68.8·10-s − 11·11-s − 18.2·12-s − 65.6·14-s + 75.4·15-s − 79.5·16-s − 18.2·17-s + 41.9·18-s − 130.·19-s − 90.3·20-s − 71.8·21-s + 39.1·22-s + 69.1·23-s − 46.2·24-s + 248.·25-s + 151.·27-s + 86.1·28-s − 92.1·29-s − 268.·30-s + ⋯ |
| L(s) = 1 | − 1.25·2-s − 0.750·3-s + 0.584·4-s − 1.72·5-s + 0.944·6-s + 0.995·7-s + 0.523·8-s − 0.436·9-s + 2.17·10-s − 0.301·11-s − 0.438·12-s − 1.25·14-s + 1.29·15-s − 1.24·16-s − 0.260·17-s + 0.549·18-s − 1.57·19-s − 1.01·20-s − 0.747·21-s + 0.379·22-s + 0.627·23-s − 0.392·24-s + 1.99·25-s + 1.07·27-s + 0.581·28-s − 0.590·29-s − 1.63·30-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 | \( 1 + 3.55T + 8T^{2} \) |
| 3 | \( 1 + 3.90T + 27T^{2} \) |
| 5 | \( 1 + 19.3T + 125T^{2} \) |
| 7 | \( 1 - 18.4T + 343T^{2} \) |
| 17 | \( 1 + 18.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 130.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 69.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 92.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 166.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 174.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 6.39T + 6.89e4T^{2} \) |
| 43 | \( 1 + 549.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 237.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 179.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 805.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 495.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 134.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 555.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 202.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 436.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.16e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.30e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 684.T + 9.12e5T^{2} \) |
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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
−8.347940611675708507077500692196, −8.053382626944653365008718740542, −7.15814805308382427171221072656, −6.42899599755047367031944865494, −4.92153282598717877049980189935, −4.64102405847883032796023464848, −3.45361335593660322255658105049, −1.99577171860582893462516457170, −0.70889498737615047366911337347, 0,
0.70889498737615047366911337347, 1.99577171860582893462516457170, 3.45361335593660322255658105049, 4.64102405847883032796023464848, 4.92153282598717877049980189935, 6.42899599755047367031944865494, 7.15814805308382427171221072656, 8.053382626944653365008718740542, 8.347940611675708507077500692196
