| L(s) = 1 | + (0.766 + 0.642i)2-s + (−2.46 + 0.897i)3-s + (0.173 + 0.984i)4-s + (−0.00995 + 0.0564i)5-s + (−2.46 − 0.897i)6-s + (−1.80 − 3.12i)7-s + (−0.500 + 0.866i)8-s + (2.97 − 2.49i)9-s + (−0.0439 + 0.0368i)10-s + (0.5 − 0.866i)11-s + (−1.31 − 2.27i)12-s + (−5.62 − 2.04i)13-s + (0.625 − 3.54i)14-s + (−0.0261 − 0.148i)15-s + (−0.939 + 0.342i)16-s + (2.74 + 2.30i)17-s + ⋯ |
| L(s) = 1 | + (0.541 + 0.454i)2-s + (−1.42 + 0.518i)3-s + (0.0868 + 0.492i)4-s + (−0.00445 + 0.0252i)5-s + (−1.00 − 0.366i)6-s + (−0.680 − 1.17i)7-s + (−0.176 + 0.306i)8-s + (0.992 − 0.832i)9-s + (−0.0138 + 0.0116i)10-s + (0.150 − 0.261i)11-s + (−0.378 − 0.656i)12-s + (−1.55 − 0.567i)13-s + (0.167 − 0.948i)14-s + (−0.00674 − 0.0382i)15-s + (−0.234 + 0.0855i)16-s + (0.665 + 0.558i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.220 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.397273 - 0.317643i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.397273 - 0.317643i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 - 0.642i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (-0.618 + 4.31i)T \) |
| good | 3 | \( 1 + (2.46 - 0.897i)T + (2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (0.00995 - 0.0564i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (1.80 + 3.12i)T + (-3.5 + 6.06i)T^{2} \) |
| 13 | \( 1 + (5.62 + 2.04i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.74 - 2.30i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (0.759 + 4.30i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-4.07 + 3.42i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (4.70 + 8.14i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 9.77T + 37T^{2} \) |
| 41 | \( 1 + (6.90 - 2.51i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (1.70 - 9.68i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (4.47 - 3.75i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.123 - 0.697i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (9.42 + 7.91i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.253 - 1.43i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (8.41 - 7.06i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.07 + 6.09i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-11.6 + 4.24i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-7.51 + 2.73i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (3.59 + 6.23i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-10.6 - 3.88i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (0.547 + 0.459i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.95903699801440422962499232333, −10.30566189608183015473480178279, −9.542405972216066310581226150343, −7.898190294376359055125650049689, −6.91619992623978674023994617849, −6.25609792094146590187991182388, −5.11220024511069300684411444918, −4.47488483541025406718313939538, −3.19180270622708382435062831341, −0.31682711667945071598835611263,
1.77846728952235041564292771810, 3.25634026217134572471404548951, 5.13620309020104033905248903529, 5.33077873894990622634912285066, 6.59335368644033766698558477366, 7.17838556918864807330470297726, 8.896897006882525289118258080751, 9.899511847248072561479581787194, 10.63761470340453253603850032626, 11.94861699746915242983097153724
