L(s) = 1 | + (−0.637 + 1.26i)2-s − 2.87i·3-s + (−1.18 − 1.61i)4-s + i·5-s + (3.62 + 1.83i)6-s − 2.68·7-s + (2.78 − 0.469i)8-s − 5.25·9-s + (−1.26 − 0.637i)10-s − 4.89·11-s + (−4.62 + 3.40i)12-s + 0.727·13-s + (1.71 − 3.38i)14-s + 2.87·15-s + (−1.18 + 3.82i)16-s + 6.34i·17-s + ⋯ |
L(s) = 1 | + (−0.451 + 0.892i)2-s − 1.65i·3-s + (−0.593 − 0.805i)4-s + 0.447i·5-s + (1.48 + 0.748i)6-s − 1.01·7-s + (0.986 − 0.166i)8-s − 1.75·9-s + (−0.399 − 0.201i)10-s − 1.47·11-s + (−1.33 + 0.983i)12-s + 0.201·13-s + (0.457 − 0.904i)14-s + 0.741·15-s + (−0.296 + 0.955i)16-s + 1.53i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 - 0.441i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0182293 + 0.0783734i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0182293 + 0.0783734i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.637 - 1.26i)T \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (-2.20 - 4.25i)T \) |
good | 3 | \( 1 + 2.87iT - 3T^{2} \) |
| 7 | \( 1 + 2.68T + 7T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 - 0.727T + 13T^{2} \) |
| 17 | \( 1 - 6.34iT - 17T^{2} \) |
| 19 | \( 1 - 2.68T + 19T^{2} \) |
| 29 | \( 1 + 4.27T + 29T^{2} \) |
| 31 | \( 1 + 6.13iT - 31T^{2} \) |
| 37 | \( 1 - 3.87iT - 37T^{2} \) |
| 41 | \( 1 + 8.07T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 - 9.80iT - 47T^{2} \) |
| 53 | \( 1 + 11.1iT - 53T^{2} \) |
| 59 | \( 1 - 1.87iT - 59T^{2} \) |
| 61 | \( 1 + 6.93iT - 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 + 7.24iT - 71T^{2} \) |
| 73 | \( 1 - 7.90T + 73T^{2} \) |
| 79 | \( 1 + 3.67T + 79T^{2} \) |
| 83 | \( 1 - 4.75T + 83T^{2} \) |
| 89 | \( 1 - 8.12iT - 89T^{2} \) |
| 97 | \( 1 - 2.68iT - 97T^{2} \) |
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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
−11.41598326762085838329689899691, −10.40291109561354958678510307835, −9.523931766894998811913125613125, −8.198780516796029493967779590314, −7.78906640387482755890890860154, −6.84252379886892267954377924890, −6.21157318663622668423845138266, −5.37534133821970313371183255947, −3.28652383827577006801886914786, −1.75842431750839318442323475547,
0.05531188581005735046006962523, 2.79589200665426229532317916033, 3.47900116940985218018202864017, 4.78132255548904087076555883419, 5.33083195941804661984752884599, 7.21021288149269082380065576749, 8.517905015283230929283246203452, 9.166605155930083334203777905207, 9.968759733441267605709911281800, 10.37639513466634669077534260991