| L(s) = 1 | − 4.00·2-s − 8.62·3-s + 8.00·4-s + 18.4·5-s + 34.4·6-s − 15.5·7-s − 0.0259·8-s + 47.3·9-s − 73.8·10-s + 11·11-s − 69.0·12-s + 62.0·14-s − 159.·15-s − 63.9·16-s + 33.9·17-s − 189.·18-s + 77.0·19-s + 147.·20-s + 133.·21-s − 44.0·22-s + 140.·23-s + 0.223·24-s + 216.·25-s − 175.·27-s − 124.·28-s + 23.2·29-s + 637.·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.65·3-s + 1.00·4-s + 1.65·5-s + 2.34·6-s − 0.837·7-s − 0.00114·8-s + 1.75·9-s − 2.33·10-s + 0.301·11-s − 1.66·12-s + 1.18·14-s − 2.74·15-s − 0.999·16-s + 0.484·17-s − 2.48·18-s + 0.930·19-s + 1.65·20-s + 1.38·21-s − 0.426·22-s + 1.27·23-s + 0.00190·24-s + 1.72·25-s − 1.25·27-s − 0.838·28-s + 0.148·29-s + 3.87·30-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 | \( 1 + 4.00T + 8T^{2} \) |
| 3 | \( 1 + 8.62T + 27T^{2} \) |
| 5 | \( 1 - 18.4T + 125T^{2} \) |
| 7 | \( 1 + 15.5T + 343T^{2} \) |
| 17 | \( 1 - 33.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 77.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 140.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 23.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 7.98T + 2.97e4T^{2} \) |
| 37 | \( 1 + 392.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 484.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 249.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 250.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 580.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 796.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 879.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 688.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 83.5T + 3.57e5T^{2} \) |
| 73 | \( 1 + 216.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.19e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 473.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 515.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 609.T + 9.12e5T^{2} \) |
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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
−8.924433691437278766446438368145, −7.55205436594439057394596647215, −6.69067463112445504342883293889, −6.37485278393219139813926075561, −5.42103783482042827568669533647, −4.87704583043293300694302559520, −3.13366176735919526759875853169, −1.68763600537760553250228433825, −1.06897972188163837481674452319, 0,
1.06897972188163837481674452319, 1.68763600537760553250228433825, 3.13366176735919526759875853169, 4.87704583043293300694302559520, 5.42103783482042827568669533647, 6.37485278393219139813926075561, 6.69067463112445504342883293889, 7.55205436594439057394596647215, 8.924433691437278766446438368145
