L(s) = 1 | + (0.841 + 1.45i)5-s + (1.65 + 2.06i)7-s + (0.622 − 1.07i)11-s + (1.96 − 3.39i)13-s + (1.62 + 2.81i)17-s + (2.36 − 4.09i)19-s + (−0.199 − 0.344i)23-s + (1.08 − 1.87i)25-s + (3.19 + 5.54i)29-s − 0.578·31-s + (−1.61 + 4.14i)35-s + (2.72 − 4.71i)37-s + (−4.20 + 7.27i)41-s + (2.46 + 4.26i)43-s + 0.425·47-s + ⋯ |
L(s) = 1 | + (0.376 + 0.651i)5-s + (0.625 + 0.780i)7-s + (0.187 − 0.325i)11-s + (0.543 − 0.941i)13-s + (0.394 + 0.683i)17-s + (0.541 − 0.938i)19-s + (−0.0415 − 0.0718i)23-s + (0.216 − 0.375i)25-s + (0.594 + 1.02i)29-s − 0.103·31-s + (−0.273 + 0.701i)35-s + (0.447 − 0.774i)37-s + (−0.656 + 1.13i)41-s + (0.375 + 0.650i)43-s + 0.0620·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.821 - 0.570i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.821 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.097811347\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.097811347\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.65 - 2.06i)T \) |
good | 5 | \( 1 + (-0.841 - 1.45i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.622 + 1.07i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.96 + 3.39i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.62 - 2.81i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.36 + 4.09i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.199 + 0.344i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.19 - 5.54i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 0.578T + 31T^{2} \) |
| 37 | \( 1 + (-2.72 + 4.71i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.20 - 7.27i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.46 - 4.26i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 0.425T + 47T^{2} \) |
| 53 | \( 1 + (-0.466 - 0.807i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 6.05T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 + 9.41T + 67T^{2} \) |
| 71 | \( 1 + 8.46T + 71T^{2} \) |
| 73 | \( 1 + (-6.82 - 11.8i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 5.53T + 79T^{2} \) |
| 83 | \( 1 + (-8.03 - 13.9i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-6.03 + 10.4i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.86 + 10.1i)T + (-48.5 + 84.0i)T^{2} \) |
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
−9.541188219719739604209715067862, −8.647481414336943698511048293988, −8.105738758154133130118184438954, −7.10204574443756724047190096551, −6.16905758234942240639379034377, −5.57120980577810520404615435995, −4.62718970043698334151724902108, −3.29378464884693829496455903532, −2.58315506230191111651358892597, −1.23584207546558709937158737906,
1.03876269315154042954789364697, 1.94706865473056276308894450581, 3.51467189149117179533233430652, 4.40612628391602668643064774022, 5.14460133086339920491063169363, 6.09535011916599297499497713679, 7.07458746685054541831359105514, 7.78257567553581508581848848736, 8.644415646015591334301233838573, 9.404633787032583426958411219839