| L(s) = 1 | + (−0.766 + 0.642i)2-s + (−0.845 + 1.51i)3-s + (0.173 − 0.984i)4-s + (−2.22 + 0.392i)5-s + (−0.324 − 1.70i)6-s + (−1.16 + 2.02i)7-s + (0.500 + 0.866i)8-s + (−1.57 − 2.55i)9-s + (1.45 − 1.73i)10-s + (−2.52 + 1.45i)11-s + (1.34 + 1.09i)12-s + (−0.451 − 1.24i)13-s + (−0.405 − 2.30i)14-s + (1.28 − 3.70i)15-s + (−0.939 − 0.342i)16-s + (3.72 + 4.43i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 + 0.454i)2-s + (−0.488 + 0.872i)3-s + (0.0868 − 0.492i)4-s + (−0.996 + 0.175i)5-s + (−0.132 − 0.694i)6-s + (−0.441 + 0.764i)7-s + (0.176 + 0.306i)8-s + (−0.523 − 0.852i)9-s + (0.459 − 0.548i)10-s + (−0.760 + 0.438i)11-s + (0.387 + 0.316i)12-s + (−0.125 − 0.344i)13-s + (−0.108 − 0.614i)14-s + (0.333 − 0.955i)15-s + (−0.234 − 0.0855i)16-s + (0.902 + 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.235i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0483156 + 0.404747i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0483156 + 0.404747i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 3 | \( 1 + (0.845 - 1.51i)T \) |
| 19 | \( 1 + (1.79 - 3.97i)T \) |
| good | 5 | \( 1 + (2.22 - 0.392i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (1.16 - 2.02i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.52 - 1.45i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.451 + 1.24i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-3.72 - 4.43i)T + (-2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-8.06 - 1.42i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (1.64 + 1.38i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-5.27 - 3.04i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2.98iT - 37T^{2} \) |
| 41 | \( 1 + (8.52 + 3.10i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.0666 + 0.377i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (6.57 - 7.83i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.494 + 2.80i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-2.53 + 2.12i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.01 + 5.77i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (10.4 - 12.4i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (2.29 + 13.0i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-5.84 - 2.12i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-1.77 + 4.87i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-1.62 - 0.938i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.82 + 2.11i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-6.13 - 7.30i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−14.72550847244788843328376990183, −12.75578698063373552552798124510, −11.86378989881312496378870521008, −10.71054738482385033987020910731, −9.930394969759937645665956761910, −8.703446914004829871752244664858, −7.67225263052151816011219147453, −6.17799612034792974010631637343, −5.04458390727069714164509891604, −3.39694726692243166288556302507,
0.56398526482336628179128257017, 3.03882022715645526080902081914, 4.88951476584706558483635810530, 6.82064320304095244994200543103, 7.57784791331851590588679257289, 8.616057752452003239627500745071, 10.15707064720957392081105709019, 11.24639305673518056276023158217, 11.85921427173253759513404279198, 13.00981653787356950046231093681
