L(s) = 1 | + 2-s + 4-s − 1.23·5-s + 7-s + 8-s − 1.23·10-s + 2·11-s + 5.23·13-s + 14-s + 16-s − 2.47·17-s − 19-s − 1.23·20-s + 2·22-s + 5.70·23-s − 3.47·25-s + 5.23·26-s + 28-s − 8.47·29-s + 7.70·31-s + 32-s − 2.47·34-s − 1.23·35-s + 0.763·37-s − 38-s − 1.23·40-s + 6.94·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.552·5-s + 0.377·7-s + 0.353·8-s − 0.390·10-s + 0.603·11-s + 1.45·13-s + 0.267·14-s + 0.250·16-s − 0.599·17-s − 0.229·19-s − 0.276·20-s + 0.426·22-s + 1.19·23-s − 0.694·25-s + 1.02·26-s + 0.188·28-s − 1.57·29-s + 1.38·31-s + 0.176·32-s − 0.423·34-s − 0.208·35-s + 0.125·37-s − 0.162·38-s − 0.195·40-s + 1.08·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.974951763\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.974951763\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 + 1.23T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 5.23T + 13T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 23 | \( 1 - 5.70T + 23T^{2} \) |
| 29 | \( 1 + 8.47T + 29T^{2} \) |
| 31 | \( 1 - 7.70T + 31T^{2} \) |
| 37 | \( 1 - 0.763T + 37T^{2} \) |
| 41 | \( 1 - 6.94T + 41T^{2} \) |
| 43 | \( 1 - 1.52T + 43T^{2} \) |
| 47 | \( 1 + 2.76T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 - 2.94T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−8.814609753080155711045642618726, −8.185768749285079129783062342287, −7.33722278593866393344235830350, −6.52772577238131241752626404274, −5.87183178401523548374110191246, −4.89105587604350137745269534823, −4.03126850180966068814978179903, −3.51981593083694646397659457822, −2.25839439048152155889053900188, −1.06431382605094169534341828836,
1.06431382605094169534341828836, 2.25839439048152155889053900188, 3.51981593083694646397659457822, 4.03126850180966068814978179903, 4.89105587604350137745269534823, 5.87183178401523548374110191246, 6.52772577238131241752626404274, 7.33722278593866393344235830350, 8.185768749285079129783062342287, 8.814609753080155711045642618726