L(s) = 1 | + 0.477·3-s − i·5-s + (2.09 − 1.61i)7-s − 2.77·9-s + 4.66i·11-s + 3.77i·13-s − 0.477i·15-s + 3.77i·17-s − 7.42·19-s + (1 − 0.772i)21-s − 3.23i·23-s − 25-s − 2.75·27-s − 3.77·29-s − 0.954·31-s + ⋯ |
L(s) = 1 | + 0.275·3-s − 0.447i·5-s + (0.791 − 0.611i)7-s − 0.924·9-s + 1.40i·11-s + 1.04i·13-s − 0.123i·15-s + 0.914i·17-s − 1.70·19-s + (0.218 − 0.168i)21-s − 0.674i·23-s − 0.200·25-s − 0.530·27-s − 0.700·29-s − 0.171·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7955926910\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7955926910\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + (-2.09 + 1.61i)T \) |
good | 3 | \( 1 - 0.477T + 3T^{2} \) |
| 11 | \( 1 - 4.66iT - 11T^{2} \) |
| 13 | \( 1 - 3.77iT - 13T^{2} \) |
| 17 | \( 1 - 3.77iT - 17T^{2} \) |
| 19 | \( 1 + 7.42T + 19T^{2} \) |
| 23 | \( 1 + 3.23iT - 23T^{2} \) |
| 29 | \( 1 + 3.77T + 29T^{2} \) |
| 31 | \( 1 + 0.954T + 31T^{2} \) |
| 37 | \( 1 + 5.54T + 37T^{2} \) |
| 41 | \( 1 + 6iT - 41T^{2} \) |
| 43 | \( 1 - 12.5iT - 43T^{2} \) |
| 47 | \( 1 + 7.89T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 9.33T + 59T^{2} \) |
| 61 | \( 1 + 13.5iT - 61T^{2} \) |
| 67 | \( 1 - 6.09iT - 67T^{2} \) |
| 71 | \( 1 - 9.33iT - 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 11.1iT - 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 - 12iT - 89T^{2} \) |
| 97 | \( 1 + 8.22iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.188309240844471515218451126558, −8.453872052701306468377177122368, −7.980989792254326292924024887522, −6.97531333696845349120802301274, −6.31803851397339619561641298316, −5.16536536468675331913516349237, −4.41388529060510590123456110720, −3.85275182712232512506725121441, −2.25966340174798885001726353107, −1.66357356660945894760831464232,
0.24033633716801940210936916783, 1.97298909574946531681187440859, 2.92479101026762195505056649634, 3.56013583478075825578220276875, 4.92312436612485950483565186665, 5.68613036626058232875567490051, 6.19327504624598658402075913779, 7.40018442606084303511725002705, 8.149814545543627085367320636995, 8.676169053181831453812335221286