| L(s) = 1 | + 2.27·2-s − 3.20·3-s + 3.19·4-s − 0.137·5-s − 7.31·6-s − 1.86·7-s + 2.72·8-s + 7.30·9-s − 0.313·10-s + 2.62·11-s − 10.2·12-s − 1.74·13-s − 4.26·14-s + 0.441·15-s − 0.171·16-s + 1.52·17-s + 16.6·18-s − 4.89·19-s − 0.440·20-s + 6.00·21-s + 5.98·22-s + 4.37·23-s − 8.76·24-s − 4.98·25-s − 3.97·26-s − 13.8·27-s − 5.97·28-s + ⋯ |
| L(s) = 1 | + 1.61·2-s − 1.85·3-s + 1.59·4-s − 0.0615·5-s − 2.98·6-s − 0.706·7-s + 0.965·8-s + 2.43·9-s − 0.0992·10-s + 0.791·11-s − 2.96·12-s − 0.484·13-s − 1.13·14-s + 0.114·15-s − 0.0429·16-s + 0.369·17-s + 3.92·18-s − 1.12·19-s − 0.0984·20-s + 1.30·21-s + 1.27·22-s + 0.911·23-s − 1.78·24-s − 0.996·25-s − 0.780·26-s − 2.65·27-s − 1.12·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 - 2.27T + 2T^{2} \) |
| 3 | \( 1 + 3.20T + 3T^{2} \) |
| 5 | \( 1 + 0.137T + 5T^{2} \) |
| 7 | \( 1 + 1.86T + 7T^{2} \) |
| 11 | \( 1 - 2.62T + 11T^{2} \) |
| 13 | \( 1 + 1.74T + 13T^{2} \) |
| 17 | \( 1 - 1.52T + 17T^{2} \) |
| 19 | \( 1 + 4.89T + 19T^{2} \) |
| 23 | \( 1 - 4.37T + 23T^{2} \) |
| 29 | \( 1 + 0.0316T + 29T^{2} \) |
| 31 | \( 1 + 2.83T + 31T^{2} \) |
| 37 | \( 1 - 0.00930T + 37T^{2} \) |
| 41 | \( 1 + 8.32T + 41T^{2} \) |
| 47 | \( 1 + 5.62T + 47T^{2} \) |
| 53 | \( 1 - 7.73T + 53T^{2} \) |
| 59 | \( 1 + 2.17T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 + 7.73T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 - 4.75T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 4.30T + 83T^{2} \) |
| 89 | \( 1 - 7.69T + 89T^{2} \) |
| 97 | \( 1 - 1.61T + 97T^{2} \) |
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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
−9.050336806915101437784303309694, −7.43215344806241052300203254422, −6.71347293456404170256964058340, −6.23934565309073307735297607350, −5.60331590375947463401108359993, −4.80136826771494524479518432937, −4.18105209589289755862437129374, −3.26499755217641187874316831594, −1.72435963223482797793316498704, 0,
1.72435963223482797793316498704, 3.26499755217641187874316831594, 4.18105209589289755862437129374, 4.80136826771494524479518432937, 5.60331590375947463401108359993, 6.23934565309073307735297607350, 6.71347293456404170256964058340, 7.43215344806241052300203254422, 9.050336806915101437784303309694
