| L(s) = 1 | − 4.18i·5-s + 2.44·7-s − 4.24i·11-s + 0.717i·13-s + 3·17-s − 6.24i·19-s − 1.01·23-s − 12.4·25-s − 4.18i·29-s − 8.36·31-s − 10.2i·35-s − 7.64i·37-s + 8.48·41-s + 12.2i·43-s − 8.36·47-s + ⋯ |
| L(s) = 1 | − 1.87i·5-s + 0.925·7-s − 1.27i·11-s + 0.198i·13-s + 0.727·17-s − 1.43i·19-s − 0.211·23-s − 2.49·25-s − 0.776i·29-s − 1.50·31-s − 1.73i·35-s − 1.25i·37-s + 1.32·41-s + 1.86i·43-s − 1.21·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.784387972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.784387972\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 4.18iT - 5T^{2} \) |
| 7 | \( 1 - 2.44T + 7T^{2} \) |
| 11 | \( 1 + 4.24iT - 11T^{2} \) |
| 13 | \( 1 - 0.717iT - 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 + 6.24iT - 19T^{2} \) |
| 23 | \( 1 + 1.01T + 23T^{2} \) |
| 29 | \( 1 + 4.18iT - 29T^{2} \) |
| 31 | \( 1 + 8.36T + 31T^{2} \) |
| 37 | \( 1 + 7.64iT - 37T^{2} \) |
| 41 | \( 1 - 8.48T + 41T^{2} \) |
| 43 | \( 1 - 12.2iT - 43T^{2} \) |
| 47 | \( 1 + 8.36T + 47T^{2} \) |
| 53 | \( 1 - 2.02iT - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 5.61iT - 61T^{2} \) |
| 67 | \( 1 + 2.24iT - 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 - 7.48T + 73T^{2} \) |
| 79 | \( 1 + 5.91T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 17.4T + 89T^{2} \) |
| 97 | \( 1 - 0.485T + 97T^{2} \) |
| 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
−8.007757279440413953611933903724, −7.49292020283424541389046201498, −6.18006046815615613730734775324, −5.57734953818446361978158101500, −4.93616989979347424299258446780, −4.37603303539028715401755237970, −3.48606098337424360934093842567, −2.20826830734749858362392708242, −1.21931002146767016907726461826, −0.48621200105789845091891855374,
1.66961549746851200580032703117, 2.22438305450245863533618121590, 3.35413837088602520160857857830, 3.84512968681007347655002653428, 4.96626443115874944355833174954, 5.68554755320229931412439161313, 6.53473766131592125431723646295, 7.17180319164275554162572885219, 7.71913097716262430203443805428, 8.210440138532979218064710020053
