L(s) = 1 | − 3.23·5-s + 3.23·7-s − 2·11-s − 13-s + 4.47·17-s + 0.763·19-s − 6.47·23-s + 5.47·25-s + 4.47·29-s − 5.70·31-s − 10.4·35-s − 8.47·37-s + 3.23·41-s − 2.47·43-s + 10.9·47-s + 3.47·49-s + 0.472·53-s + 6.47·55-s + 0.472·59-s − 3.52·61-s + 3.23·65-s + 11.2·67-s + 4.47·71-s − 8.47·73-s − 6.47·77-s + 8.94·79-s − 16.4·83-s + ⋯ |
L(s) = 1 | − 1.44·5-s + 1.22·7-s − 0.603·11-s − 0.277·13-s + 1.08·17-s + 0.175·19-s − 1.34·23-s + 1.09·25-s + 0.830·29-s − 1.02·31-s − 1.77·35-s − 1.39·37-s + 0.505·41-s − 0.376·43-s + 1.59·47-s + 0.496·49-s + 0.0648·53-s + 0.872·55-s + 0.0614·59-s − 0.451·61-s + 0.401·65-s + 1.37·67-s + 0.530·71-s − 0.991·73-s − 0.737·77-s + 1.00·79-s − 1.80·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 + 3.23T + 5T^{2} \) |
| 7 | \( 1 - 3.23T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 0.763T + 19T^{2} \) |
| 23 | \( 1 + 6.47T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 5.70T + 31T^{2} \) |
| 37 | \( 1 + 8.47T + 37T^{2} \) |
| 41 | \( 1 - 3.23T + 41T^{2} \) |
| 43 | \( 1 + 2.47T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 - 0.472T + 53T^{2} \) |
| 59 | \( 1 - 0.472T + 59T^{2} \) |
| 61 | \( 1 + 3.52T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 - 4.47T + 71T^{2} \) |
| 73 | \( 1 + 8.47T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 - 4.47T + 97T^{2} \) |
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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
−8.114113798361225234900045230478, −7.57871733972341701049807863213, −6.98471816495376986095750211557, −5.67129765641423320932423506944, −5.09984733621914876919048497448, −4.22621961411773029635001050825, −3.63497509535695099879959711468, −2.54382161355912910327164406839, −1.35216872924020757910259893925, 0,
1.35216872924020757910259893925, 2.54382161355912910327164406839, 3.63497509535695099879959711468, 4.22621961411773029635001050825, 5.09984733621914876919048497448, 5.67129765641423320932423506944, 6.98471816495376986095750211557, 7.57871733972341701049807863213, 8.114113798361225234900045230478