| L(s) = 1 | + (0.376 + 1.84i)2-s + (0.987 + 0.160i)3-s + (−1.41 + 0.603i)4-s + (−0.252 + 0.0623i)5-s + (0.0757 + 1.87i)6-s + (2.18 − 4.59i)7-s + (0.491 + 0.712i)8-s + (0.948 + 0.316i)9-s + (−0.210 − 0.442i)10-s + (5.06 − 1.69i)11-s + (−1.49 + 0.368i)12-s + (−3.01 + 1.97i)13-s + (9.29 + 2.29i)14-s + (−0.259 + 0.0209i)15-s + (−3.26 + 3.39i)16-s + (2.09 − 4.41i)17-s + ⋯ |
| L(s) = 1 | + (0.266 + 1.30i)2-s + (0.569 + 0.0926i)3-s + (−0.708 + 0.301i)4-s + (−0.113 + 0.0278i)5-s + (0.0309 + 0.767i)6-s + (0.824 − 1.73i)7-s + (0.173 + 0.251i)8-s + (0.316 + 0.105i)9-s + (−0.0664 − 0.139i)10-s + (1.52 − 0.509i)11-s + (−0.431 + 0.106i)12-s + (−0.836 + 0.547i)13-s + (2.48 + 0.612i)14-s + (−0.0670 + 0.00541i)15-s + (−0.815 + 0.848i)16-s + (0.508 − 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.284 - 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.79307 + 1.33806i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.79307 + 1.33806i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.987 - 0.160i)T \) |
| 13 | \( 1 + (3.01 - 1.97i)T \) |
| good | 2 | \( 1 + (-0.376 - 1.84i)T + (-1.83 + 0.783i)T^{2} \) |
| 5 | \( 1 + (0.252 - 0.0623i)T + (4.42 - 2.32i)T^{2} \) |
| 7 | \( 1 + (-2.18 + 4.59i)T + (-4.42 - 5.42i)T^{2} \) |
| 11 | \( 1 + (-5.06 + 1.69i)T + (8.79 - 6.60i)T^{2} \) |
| 17 | \( 1 + (-2.09 + 4.41i)T + (-10.7 - 13.1i)T^{2} \) |
| 19 | \( 1 + (-0.801 - 1.38i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.14 - 7.18i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.918 + 4.49i)T + (-26.6 + 11.3i)T^{2} \) |
| 31 | \( 1 + (4.88 + 2.56i)T + (17.6 + 25.5i)T^{2} \) |
| 37 | \( 1 + (4.10 - 2.59i)T + (15.8 - 33.4i)T^{2} \) |
| 41 | \( 1 + (-4.16 - 0.676i)T + (38.8 + 12.9i)T^{2} \) |
| 43 | \( 1 + (-4.44 - 2.80i)T + (18.4 + 38.8i)T^{2} \) |
| 47 | \( 1 + (-0.976 - 8.04i)T + (-45.6 + 11.2i)T^{2} \) |
| 53 | \( 1 + (4.15 + 6.01i)T + (-18.7 + 49.5i)T^{2} \) |
| 59 | \( 1 + (-4.63 - 4.82i)T + (-2.37 + 58.9i)T^{2} \) |
| 61 | \( 1 + (-8.02 - 0.647i)T + (60.2 + 9.78i)T^{2} \) |
| 67 | \( 1 + (10.6 + 4.53i)T + (46.4 + 48.3i)T^{2} \) |
| 71 | \( 1 + (5.26 - 6.44i)T + (-14.2 - 69.5i)T^{2} \) |
| 73 | \( 1 + (-3.31 - 2.93i)T + (8.79 + 72.4i)T^{2} \) |
| 79 | \( 1 + (0.587 + 4.83i)T + (-76.7 + 18.9i)T^{2} \) |
| 83 | \( 1 + (-0.722 + 1.90i)T + (-62.1 - 55.0i)T^{2} \) |
| 89 | \( 1 + (5.07 - 8.79i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.551 + 1.90i)T + (-81.9 + 51.8i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.28271277968279983193130019006, −9.912500256065772721771326036761, −9.208552816268172624108267378119, −7.80852304943475024643477700816, −7.58117928385809982933462185003, −6.78114915214775942958056875164, −5.56917129700554844692838683566, −4.36249962303826906682926330257, −3.78448103562906521077179852915, −1.56912522786000559524200986956,
1.73180269318982051478208522223, 2.39571161640783225936257137515, 3.66708901884415500271463442763, 4.65961941358128075367959338198, 5.87521172848960725851817293145, 7.20687714183352494623277938194, 8.395563334574004010947606992019, 9.066965493574693430203308715160, 9.914577336278459135787112015624, 10.85955355634206848533464446779
