| L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s − 3.52i·7-s − i·8-s − 9-s + 5.25·11-s + i·12-s − 0.619i·13-s + 3.52·14-s + 16-s + 3.44i·17-s − i·18-s − 2.27·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 1.33i·7-s − 0.353i·8-s − 0.333·9-s + 1.58·11-s + 0.288i·12-s − 0.171i·13-s + 0.941·14-s + 0.250·16-s + 0.835i·17-s − 0.235i·18-s − 0.521·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.969382753\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.969382753\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 3.52iT - 7T^{2} \) |
| 11 | \( 1 - 5.25T + 11T^{2} \) |
| 13 | \( 1 + 0.619iT - 13T^{2} \) |
| 17 | \( 1 - 3.44iT - 17T^{2} \) |
| 19 | \( 1 + 2.27T + 19T^{2} \) |
| 23 | \( 1 - 8.95iT - 23T^{2} \) |
| 29 | \( 1 - 2.68T + 29T^{2} \) |
| 31 | \( 1 - 9.76T + 31T^{2} \) |
| 37 | \( 1 + 1.00iT - 37T^{2} \) |
| 41 | \( 1 - 6.29T + 41T^{2} \) |
| 43 | \( 1 - 1.51iT - 43T^{2} \) |
| 47 | \( 1 - 10.6iT - 47T^{2} \) |
| 53 | \( 1 + 0.553iT - 53T^{2} \) |
| 59 | \( 1 + 0.278T + 59T^{2} \) |
| 61 | \( 1 - 3.68T + 61T^{2} \) |
| 67 | \( 1 + 11.9iT - 67T^{2} \) |
| 71 | \( 1 - 4.74T + 71T^{2} \) |
| 73 | \( 1 + 9.83iT - 73T^{2} \) |
| 79 | \( 1 + 6.52T + 79T^{2} \) |
| 83 | \( 1 + 2.65iT - 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 - 4.37iT - 97T^{2} \) |
| 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
−8.220678836316453848535715187634, −7.72809029040775880816209493082, −6.99150521676149734788720082664, −6.40276386783590927986864166833, −5.87366902568790564106815220884, −4.60222448928983777156473461316, −4.03617471531341943503828506104, −3.23325242106104359897292377856, −1.60502955011433113408841793821, −0.859138679468406913755472085560,
0.866803226892878690149120954925, 2.28112497982379251635234039446, 2.79101944733761079419077324391, 3.94958161541335306519994054689, 4.53188371680276134809194198859, 5.34058917657564539468722884927, 6.28704813646933295039651673062, 6.77704358601125118510220249512, 8.241348359859037708152467550105, 8.742580833097153224417646750782
