| L(s) = 1 | + (−1.57 + 0.711i)3-s + (2.16 + 0.559i)5-s + (1.98 − 2.24i)9-s − 2.19i·11-s − 5.07·13-s + (−3.81 + 0.657i)15-s − 2.90i·17-s − 0.155i·19-s − 3.63·23-s + (4.37 + 2.42i)25-s + (−1.53 + 4.96i)27-s + 3.88i·29-s + 9.79i·31-s + (1.56 + 3.47i)33-s − 4.01i·37-s + ⋯ |
| L(s) = 1 | + (−0.911 + 0.411i)3-s + (0.968 + 0.250i)5-s + (0.662 − 0.749i)9-s − 0.662i·11-s − 1.40·13-s + (−0.985 + 0.169i)15-s − 0.705i·17-s − 0.0355i·19-s − 0.757·23-s + (0.874 + 0.484i)25-s + (−0.295 + 0.955i)27-s + 0.722i·29-s + 1.75i·31-s + (0.272 + 0.604i)33-s − 0.659i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.321 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.321 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9814818002\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9814818002\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.57 - 0.711i)T \) |
| 5 | \( 1 + (-2.16 - 0.559i)T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2.19iT - 11T^{2} \) |
| 13 | \( 1 + 5.07T + 13T^{2} \) |
| 17 | \( 1 + 2.90iT - 17T^{2} \) |
| 19 | \( 1 + 0.155iT - 19T^{2} \) |
| 23 | \( 1 + 3.63T + 23T^{2} \) |
| 29 | \( 1 - 3.88iT - 29T^{2} \) |
| 31 | \( 1 - 9.79iT - 31T^{2} \) |
| 37 | \( 1 + 4.01iT - 37T^{2} \) |
| 41 | \( 1 + 7.97T + 41T^{2} \) |
| 43 | \( 1 - 2.59iT - 43T^{2} \) |
| 47 | \( 1 - 12.2iT - 47T^{2} \) |
| 53 | \( 1 - 9.42T + 53T^{2} \) |
| 59 | \( 1 - 6.80T + 59T^{2} \) |
| 61 | \( 1 - 6.97iT - 61T^{2} \) |
| 67 | \( 1 - 6.51iT - 67T^{2} \) |
| 71 | \( 1 - 2.22iT - 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 + 3.68T + 79T^{2} \) |
| 83 | \( 1 - 13.6iT - 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 - 1.73T + 97T^{2} \) |
| 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
−9.287137732521058550382848183715, −8.314545714743812238865032114516, −7.06142932150427922832851098681, −6.80193448806898361313740601100, −5.71452533953834651469040522375, −5.29328058708168654734734562212, −4.53857263766852526628242213395, −3.35181264231236850237507872878, −2.41647762448206316011193814772, −1.12786813051718906363599048751,
0.36973715439422729345646606987, 1.88256083426223826163637159130, 2.29607741131210284044687296866, 3.98723042542056108222645862403, 4.88590985226101056623907721079, 5.44148868314887784881074205306, 6.24004677594089719979469757891, 6.87084299350436262405813389521, 7.65912584291512038983594303399, 8.415036093834938133968401645346
