L(s) = 1 | + (−0.466 − 0.413i)2-s + (−0.345 + 0.911i)3-s + (−0.194 − 1.60i)4-s + (0.970 + 0.239i)5-s + (0.537 − 0.282i)6-s + (−0.0247 + 0.0359i)7-s + (−1.27 + 1.85i)8-s + (1.53 + 1.35i)9-s + (−0.353 − 0.512i)10-s + (0.645 − 0.572i)11-s + (1.52 + 0.375i)12-s + (3.60 + 0.102i)13-s + (0.0263 − 0.00650i)14-s + (−0.553 + 0.801i)15-s + (−1.76 + 0.436i)16-s + (−3.15 + 4.57i)17-s + ⋯ |
L(s) = 1 | + (−0.329 − 0.292i)2-s + (−0.199 + 0.525i)3-s + (−0.0971 − 0.800i)4-s + (0.434 + 0.107i)5-s + (0.219 − 0.115i)6-s + (−0.00937 + 0.0135i)7-s + (−0.451 + 0.654i)8-s + (0.511 + 0.453i)9-s + (−0.111 − 0.162i)10-s + (0.194 − 0.172i)11-s + (0.440 + 0.108i)12-s + (0.999 + 0.0285i)13-s + (0.00705 − 0.00173i)14-s + (−0.142 + 0.207i)15-s + (−0.442 + 0.109i)16-s + (−0.765 + 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.219i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.31850 + 0.146788i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.31850 + 0.146788i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.970 - 0.239i)T \) |
| 13 | \( 1 + (-3.60 - 0.102i)T \) |
good | 2 | \( 1 + (0.466 + 0.413i)T + (0.241 + 1.98i)T^{2} \) |
| 3 | \( 1 + (0.345 - 0.911i)T + (-2.24 - 1.98i)T^{2} \) |
| 7 | \( 1 + (0.0247 - 0.0359i)T + (-2.48 - 6.54i)T^{2} \) |
| 11 | \( 1 + (-0.645 + 0.572i)T + (1.32 - 10.9i)T^{2} \) |
| 17 | \( 1 + (3.15 - 4.57i)T + (-6.02 - 15.8i)T^{2} \) |
| 19 | \( 1 + 2.45T + 19T^{2} \) |
| 23 | \( 1 - 4.88T + 23T^{2} \) |
| 29 | \( 1 + (-0.577 - 0.511i)T + (3.49 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-2.50 + 1.31i)T + (17.6 - 25.5i)T^{2} \) |
| 37 | \( 1 + (-8.93 + 4.68i)T + (21.0 - 30.4i)T^{2} \) |
| 41 | \( 1 + (-1.22 + 3.23i)T + (-30.6 - 27.1i)T^{2} \) |
| 43 | \( 1 + (-4.97 - 2.61i)T + (24.4 + 35.3i)T^{2} \) |
| 47 | \( 1 + (1.09 - 9.05i)T + (-45.6 - 11.2i)T^{2} \) |
| 53 | \( 1 + (-1.90 + 2.76i)T + (-18.7 - 49.5i)T^{2} \) |
| 59 | \( 1 + (-1.97 - 0.487i)T + (52.2 + 27.4i)T^{2} \) |
| 61 | \( 1 + (6.02 + 8.72i)T + (-21.6 + 57.0i)T^{2} \) |
| 67 | \( 1 + (1.59 - 13.1i)T + (-65.0 - 16.0i)T^{2} \) |
| 71 | \( 1 + (-0.586 + 1.54i)T + (-53.1 - 47.0i)T^{2} \) |
| 73 | \( 1 + (-9.70 + 8.59i)T + (8.79 - 72.4i)T^{2} \) |
| 79 | \( 1 + (-1.59 + 13.1i)T + (-76.7 - 18.9i)T^{2} \) |
| 83 | \( 1 + (-3.98 - 10.5i)T + (-62.1 + 55.0i)T^{2} \) |
| 89 | \( 1 - 4.58T + 89T^{2} \) |
| 97 | \( 1 + (-0.804 + 0.198i)T + (85.8 - 45.0i)T^{2} \) |
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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
−10.42788689973702651485921086225, −9.375105452868978524924028210580, −8.932951810278256046041970389570, −7.83826427399254377051139847859, −6.43269253628708092653544964833, −5.95401433335362064767540312208, −4.84306264546500823714552446494, −3.99831666574837751367247704198, −2.39176469877581449622220902059, −1.24282022686611987946039869271,
0.904238825146340922814372717082, 2.50875806135483450910682003724, 3.76458521046221977888796641645, 4.75685358020705632641516379050, 6.20219550013116480335621892864, 6.76060354623192430227399963681, 7.49587982300174334887741794703, 8.545489249949328707958719904279, 9.138135128092646553429165698119, 9.936133877376944609337235989727