| L(s) = 1 | + i·2-s − 4-s + i·7-s − i·8-s + 11-s + 2i·13-s − 14-s + 16-s − 3i·17-s + 7·19-s + i·22-s + 3i·23-s − 2·26-s − i·28-s + 6·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.377i·7-s − 0.353i·8-s + 0.301·11-s + 0.554i·13-s − 0.267·14-s + 0.250·16-s − 0.727i·17-s + 1.60·19-s + 0.213i·22-s + 0.625i·23-s − 0.392·26-s − 0.188i·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.909425094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.909425094\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - iT - 7T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 - 3iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 5iT - 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 9iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 3T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 10iT - 67T^{2} \) |
| 71 | \( 1 - 15T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - iT - 97T^{2} \) |
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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
−8.383579310574380903308431456078, −7.43179010638705005433525106632, −7.09563296030221162340995842193, −6.24768429843713053753006092407, −5.40628817619701120899986390571, −4.98788593364171398718621805232, −3.92459631912283372037028196747, −3.19076170618081092566149823074, −2.04551820489208623065575997147, −0.806083149244069160632374182829,
0.76996385763526595083701929287, 1.61786436442582268866032541988, 2.85630377806589652700132293847, 3.40825735217003586975017461447, 4.35506229684975831408360491672, 5.01693997526559778095849496096, 5.90826727350039282342167228406, 6.63812712467642529650325551633, 7.62939488467363477864437622921, 8.078977659382167150881256894674
