| L(s) = 1 | − 1.43·2-s + 0.655·3-s + 0.0581·4-s − 1.45·5-s − 0.940·6-s − 3.21·7-s + 2.78·8-s − 2.57·9-s + 2.08·10-s − 11-s + 0.0381·12-s + 4.61·14-s − 0.954·15-s − 4.11·16-s + 2.77·17-s + 3.68·18-s − 6.77·19-s − 0.0846·20-s − 2.10·21-s + 1.43·22-s + 7.11·23-s + 1.82·24-s − 2.87·25-s − 3.65·27-s − 0.187·28-s − 3.39·29-s + 1.36·30-s + ⋯ |
| L(s) = 1 | − 1.01·2-s + 0.378·3-s + 0.0290·4-s − 0.651·5-s − 0.383·6-s − 1.21·7-s + 0.984·8-s − 0.856·9-s + 0.660·10-s − 0.301·11-s + 0.0110·12-s + 1.23·14-s − 0.246·15-s − 1.02·16-s + 0.673·17-s + 0.869·18-s − 1.55·19-s − 0.0189·20-s − 0.460·21-s + 0.305·22-s + 1.48·23-s + 0.372·24-s − 0.575·25-s − 0.702·27-s − 0.0353·28-s − 0.630·29-s + 0.249·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3740666930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3740666930\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.43T + 2T^{2} \) |
| 3 | \( 1 - 0.655T + 3T^{2} \) |
| 5 | \( 1 + 1.45T + 5T^{2} \) |
| 7 | \( 1 + 3.21T + 7T^{2} \) |
| 17 | \( 1 - 2.77T + 17T^{2} \) |
| 19 | \( 1 + 6.77T + 19T^{2} \) |
| 23 | \( 1 - 7.11T + 23T^{2} \) |
| 29 | \( 1 + 3.39T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 - 2.07T + 37T^{2} \) |
| 41 | \( 1 + 3.73T + 41T^{2} \) |
| 43 | \( 1 - 5.99T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 1.46T + 53T^{2} \) |
| 59 | \( 1 + 9.94T + 59T^{2} \) |
| 61 | \( 1 - 2.55T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 + 9.44T + 71T^{2} \) |
| 73 | \( 1 - 7.67T + 73T^{2} \) |
| 79 | \( 1 - 8.09T + 79T^{2} \) |
| 83 | \( 1 - 2.39T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 - 9.77T + 97T^{2} \) |
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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
−9.136188536371885942669693119667, −8.649511277766176742288314335819, −7.77480248281428357438640473337, −7.24900146797088029813089178054, −6.21053933663589459467469234996, −5.25630669582727562716915817060, −4.01982125675706721244525759495, −3.30935313648500200460063920225, −2.16837056251692553903035791744, −0.45044950705488336794929321386,
0.45044950705488336794929321386, 2.16837056251692553903035791744, 3.30935313648500200460063920225, 4.01982125675706721244525759495, 5.25630669582727562716915817060, 6.21053933663589459467469234996, 7.24900146797088029813089178054, 7.77480248281428357438640473337, 8.649511277766176742288314335819, 9.136188536371885942669693119667
