| L(s) = 1 | + 2·4-s + i·7-s − 3·11-s + i·13-s + 4·16-s + 3i·17-s + 4·19-s − 9i·23-s + 2i·28-s + 6·29-s + 2·31-s + i·37-s − 3·41-s + 2i·43-s − 6·44-s + ⋯ |
| L(s) = 1 | + 4-s + 0.377i·7-s − 0.904·11-s + 0.277i·13-s + 16-s + 0.727i·17-s + 0.917·19-s − 1.87i·23-s + 0.377i·28-s + 1.11·29-s + 0.359·31-s + 0.164i·37-s − 0.468·41-s + 0.304i·43-s − 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.371418260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.371418260\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - iT \) |
| good | 2 | \( 1 - 2T^{2} \) |
| 7 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 9iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - iT - 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 9iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 9T + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 - 7T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 97 | \( 1 + 5iT - 97T^{2} \) |
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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
−8.509021140382467387835576858668, −8.183246905551827455290125212617, −7.21116725326482928890417389201, −6.57491572446149152971830754892, −5.84614422203121001525573684528, −5.06587779057835608137621377311, −4.05073218470238055137998794026, −2.82586830360702652945555636247, −2.40728830464588078321785398906, −1.08309753007378548757387807657,
0.857910913631849472874066882781, 2.08008494538240381092539366279, 3.00807082063165288407293863536, 3.69837507387096144843993663812, 5.12960737531167009541858070152, 5.48896371185527099656179700822, 6.58629373050292881625933106129, 7.26262959787466682243617802239, 7.75530058644229333108313063361, 8.527239826750668076739352976436
