| L(s) = 1 | − 5-s + 1.46i·7-s − 0.486·11-s + 5.34·13-s − 2.25·17-s + 6.43i·19-s + (1.32 + 4.60i)23-s + 25-s − 3.84i·29-s − 6.74·31-s − 1.46i·35-s − 1.26i·37-s − 4.64i·41-s + 7.19i·43-s + 5.27i·47-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.552i·7-s − 0.146·11-s + 1.48·13-s − 0.547·17-s + 1.47i·19-s + (0.275 + 0.961i)23-s + 0.200·25-s − 0.713i·29-s − 1.21·31-s − 0.247i·35-s − 0.207i·37-s − 0.725i·41-s + 1.09i·43-s + 0.768i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.049304121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.049304121\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + (-1.32 - 4.60i)T \) |
| good | 7 | \( 1 - 1.46iT - 7T^{2} \) |
| 11 | \( 1 + 0.486T + 11T^{2} \) |
| 13 | \( 1 - 5.34T + 13T^{2} \) |
| 17 | \( 1 + 2.25T + 17T^{2} \) |
| 19 | \( 1 - 6.43iT - 19T^{2} \) |
| 29 | \( 1 + 3.84iT - 29T^{2} \) |
| 31 | \( 1 + 6.74T + 31T^{2} \) |
| 37 | \( 1 + 1.26iT - 37T^{2} \) |
| 41 | \( 1 + 4.64iT - 41T^{2} \) |
| 43 | \( 1 - 7.19iT - 43T^{2} \) |
| 47 | \( 1 - 5.27iT - 47T^{2} \) |
| 53 | \( 1 + 5.75T + 53T^{2} \) |
| 59 | \( 1 + 12.4iT - 59T^{2} \) |
| 61 | \( 1 - 9.62iT - 61T^{2} \) |
| 67 | \( 1 - 6.89iT - 67T^{2} \) |
| 71 | \( 1 - 8.76iT - 71T^{2} \) |
| 73 | \( 1 - 1.29T + 73T^{2} \) |
| 79 | \( 1 + 6.71iT - 79T^{2} \) |
| 83 | \( 1 + 5.08T + 83T^{2} \) |
| 89 | \( 1 - 4.45T + 89T^{2} \) |
| 97 | \( 1 + 0.599iT - 97T^{2} \) |
| 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
−8.076019761597681377324908618282, −7.51581580867526669740419147428, −6.65970603924922212040495752798, −5.83856264107267754397393257824, −5.57574674794740806777347617304, −4.41769597685046285373033123802, −3.77737406997694040321057227418, −3.15432188735106341533826602830, −2.03876475900542301547079838724, −1.21635045587026496451156411807,
0.26536126663945923036634626729, 1.23056976217628252805754071341, 2.36048621898048128394800306479, 3.31292861132918887098861013272, 3.93017323755059180228039257198, 4.68211403049074682672673553326, 5.35442720632435323166387039585, 6.38077261490997698387177158449, 6.80390431372010469458388933140, 7.48871087871841937830711328189
