| L(s) = 1 | − 2.44i·5-s + (1 + 2.44i)7-s − 2.44i·11-s + 4.89i·13-s + 2.44i·17-s − 2·19-s + 7.34i·23-s − 0.999·25-s + 6·29-s − 8·31-s + (5.99 − 2.44i)35-s − 4·37-s − 7.34i·41-s − 4.89i·43-s − 12·47-s + ⋯ |
| L(s) = 1 | − 1.09i·5-s + (0.377 + 0.925i)7-s − 0.738i·11-s + 1.35i·13-s + 0.594i·17-s − 0.458·19-s + 1.53i·23-s − 0.199·25-s + 1.11·29-s − 1.43·31-s + (1.01 − 0.414i)35-s − 0.657·37-s − 1.14i·41-s − 0.747i·43-s − 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.377 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9982400319\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9982400319\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1 - 2.44i)T \) |
| good | 5 | \( 1 + 2.44iT - 5T^{2} \) |
| 11 | \( 1 + 2.44iT - 11T^{2} \) |
| 13 | \( 1 - 4.89iT - 13T^{2} \) |
| 17 | \( 1 - 2.44iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 7.34iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 7.34iT - 41T^{2} \) |
| 43 | \( 1 + 4.89iT - 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 12.2iT - 71T^{2} \) |
| 73 | \( 1 - 14.6iT - 73T^{2} \) |
| 79 | \( 1 - 9.79iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 12.2iT - 89T^{2} \) |
| 97 | \( 1 + 4.89iT - 97T^{2} \) |
| 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
−8.708593713111611465755588759321, −8.232924760932119366334810666706, −7.23593100646273475237048327011, −6.35218317129950919295603072176, −5.57869474226415626388316960054, −5.03479101598145022740793529849, −4.18896662749837458752363235920, −3.32900047416506347700725316131, −2.01755300674500948562149542824, −1.38814178579189671761878256703,
0.27610689699380495329889963903, 1.67874550249301795228447373847, 2.84277589082081442105851650956, 3.36375950647403151529675775088, 4.56707700992666004285186064587, 4.98660763950929403122416588415, 6.36346463538286428002508950710, 6.62846549715368849129276923782, 7.64945118094453465598096252766, 7.86256143244287086622108061194
