| L(s) = 1 | − 0.618·2-s + 3-s − 1.61·4-s + 0.236·5-s − 0.618·6-s − 7-s + 2.23·8-s + 9-s − 0.145·10-s − 1.61·12-s + 1.76·13-s + 0.618·14-s + 0.236·15-s + 1.85·16-s + 4.47·17-s − 0.618·18-s + 3·19-s − 0.381·20-s − 21-s − 0.472·23-s + 2.23·24-s − 4.94·25-s − 1.09·26-s + 27-s + 1.61·28-s − 3·29-s − 0.145·30-s + ⋯ |
| L(s) = 1 | − 0.437·2-s + 0.577·3-s − 0.809·4-s + 0.105·5-s − 0.252·6-s − 0.377·7-s + 0.790·8-s + 0.333·9-s − 0.0461·10-s − 0.467·12-s + 0.489·13-s + 0.165·14-s + 0.0609·15-s + 0.463·16-s + 1.08·17-s − 0.145·18-s + 0.688·19-s − 0.0854·20-s − 0.218·21-s − 0.0984·23-s + 0.456·24-s − 0.988·25-s − 0.213·26-s + 0.192·27-s + 0.305·28-s − 0.557·29-s − 0.0266·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.451273780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.451273780\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 5 | \( 1 - 0.236T + 5T^{2} \) |
| 13 | \( 1 - 1.76T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 + 0.472T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 5.47T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 + 6.94T + 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 + 8.94T + 53T^{2} \) |
| 59 | \( 1 - 7.47T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 4.70T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 7.76T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 4.47T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.988114701860162842964666795498, −8.091215445843681269208863595386, −7.75286598807876883330113328563, −6.74894279595652241945681546692, −5.69993419467715303090900653717, −5.00709556257650255379608548006, −3.82500538898727930438318084747, −3.39825446398542176220309226827, −1.98827768583556639330875158522, −0.821788043614078500906236176220,
0.821788043614078500906236176220, 1.98827768583556639330875158522, 3.39825446398542176220309226827, 3.82500538898727930438318084747, 5.00709556257650255379608548006, 5.69993419467715303090900653717, 6.74894279595652241945681546692, 7.75286598807876883330113328563, 8.091215445843681269208863595386, 8.988114701860162842964666795498
