| L(s) = 1 | + i·2-s − 4-s + 4.16i·5-s + 3.16i·7-s − i·8-s − 4.16·10-s + 1.16i·11-s − 3.16·14-s + 16-s − 3·17-s + 5.16i·19-s − 4.16i·20-s − 1.16·22-s + 7.16·23-s − 12.3·25-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 1.86i·5-s + 1.19i·7-s − 0.353i·8-s − 1.31·10-s + 0.350i·11-s − 0.845·14-s + 0.250·16-s − 0.727·17-s + 1.18i·19-s − 0.930i·20-s − 0.247·22-s + 1.49·23-s − 2.46·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.383598126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.383598126\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 4.16iT - 5T^{2} \) |
| 7 | \( 1 - 3.16iT - 7T^{2} \) |
| 11 | \( 1 - 1.16iT - 11T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 5.16iT - 19T^{2} \) |
| 23 | \( 1 - 7.16T + 23T^{2} \) |
| 29 | \( 1 - 1.83T + 29T^{2} \) |
| 31 | \( 1 - 6.32iT - 31T^{2} \) |
| 37 | \( 1 - 3.83iT - 37T^{2} \) |
| 41 | \( 1 - 3iT - 41T^{2} \) |
| 43 | \( 1 + 9.16T + 43T^{2} \) |
| 47 | \( 1 + 4.83iT - 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 - 2.32iT - 59T^{2} \) |
| 61 | \( 1 - 0.162T + 61T^{2} \) |
| 67 | \( 1 - 2.83iT - 67T^{2} \) |
| 71 | \( 1 + 7.16iT - 71T^{2} \) |
| 73 | \( 1 + iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 3.48iT - 83T^{2} \) |
| 89 | \( 1 + 12iT - 89T^{2} \) |
| 97 | \( 1 - 4iT - 97T^{2} \) |
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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
−8.985889271855290183258430419926, −8.468221600424206788453945474113, −7.48206544829120331622863125286, −6.88388895300554691852310015640, −6.34422669474947314613923892624, −5.62515095026096038634355079079, −4.74505470485679174039870329094, −3.51215573117507194777075001124, −2.87593757874919361132471624775, −1.90758362956737602704678765056,
0.50797733738656505315827921945, 1.04472953337069469802583358714, 2.26504219719206375612137630913, 3.56581391805259495483989899682, 4.39501789588501182657036299330, 4.82076581048803638247088433968, 5.63156065801220812927831585499, 6.82596468321344723290936503086, 7.62017320920596990702862288521, 8.554823248140530528665275456768
