L(s) = 1 | + (2.20 + 0.590i)3-s + (−0.352 + 2.20i)5-s + (−1.22 + 2.34i)7-s + (1.90 + 1.10i)9-s + (−0.0644 − 0.111i)11-s + (−0.748 + 0.748i)13-s + (−2.07 + 4.65i)15-s + (0.613 − 2.28i)17-s + (3.81 − 6.59i)19-s + (−4.08 + 4.44i)21-s + (4.11 − 1.10i)23-s + (−4.75 − 1.55i)25-s + (−1.28 − 1.28i)27-s + 0.163i·29-s + (8.43 − 4.86i)31-s + ⋯ |
L(s) = 1 | + (1.27 + 0.340i)3-s + (−0.157 + 0.987i)5-s + (−0.462 + 0.886i)7-s + (0.636 + 0.367i)9-s + (−0.0194 − 0.0336i)11-s + (−0.207 + 0.207i)13-s + (−0.536 + 1.20i)15-s + (0.148 − 0.555i)17-s + (0.874 − 1.51i)19-s + (−0.890 + 0.970i)21-s + (0.857 − 0.229i)23-s + (−0.950 − 0.310i)25-s + (−0.246 − 0.246i)27-s + 0.0303i·29-s + (1.51 − 0.874i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.509 - 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.509 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.50860 + 0.860428i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.50860 + 0.860428i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.352 - 2.20i)T \) |
| 7 | \( 1 + (1.22 - 2.34i)T \) |
good | 3 | \( 1 + (-2.20 - 0.590i)T + (2.59 + 1.5i)T^{2} \) |
| 11 | \( 1 + (0.0644 + 0.111i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.748 - 0.748i)T - 13iT^{2} \) |
| 17 | \( 1 + (-0.613 + 2.28i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-3.81 + 6.59i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.11 + 1.10i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 0.163iT - 29T^{2} \) |
| 31 | \( 1 + (-8.43 + 4.86i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.0241 - 0.0900i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 11.9iT - 41T^{2} \) |
| 43 | \( 1 + (-2.76 - 2.76i)T + 43iT^{2} \) |
| 47 | \( 1 + (-2.23 + 0.597i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (1.96 - 7.32i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (4.13 + 7.16i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (11.5 + 6.64i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.84 - 1.83i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 9.50T + 71T^{2} \) |
| 73 | \( 1 + (-1.80 - 0.483i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (8.48 + 4.89i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (8.43 - 8.43i)T - 83iT^{2} \) |
| 89 | \( 1 + (-3.37 + 5.85i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (9.61 + 9.61i)T + 97iT^{2} \) |
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
−11.88006716018259112296996140584, −11.11299320114832926035824214555, −9.718499988618507525310607328269, −9.347684081624242527928018104648, −8.253951283037114931640341171253, −7.26160784352457956359102690368, −6.20169658071746501631726612931, −4.63050878412770012056355237863, −3.01263511343626423871840887116, −2.71962460792157954001303201518,
1.38797649298996749552450063165, 3.15640726948108066072566763850, 4.14791103715188574035231093505, 5.60286669724159813508069081955, 7.16697705471609774208654741703, 7.932319550537232670681947351351, 8.716401097728909228941609675587, 9.642669948998939683269623902628, 10.51592764340469998122152728321, 12.05889150124973125141119846062