L(s) = 1 | + (1.06 − 1.06i)2-s + (−0.0150 + 1.73i)3-s − 0.285i·4-s + (−2.03 − 0.933i)5-s + (1.83 + 1.86i)6-s + (1.83 + 1.83i)8-s + (−2.99 − 0.0520i)9-s + (−3.16 + 1.17i)10-s + 0.914i·11-s + (0.493 + 0.00428i)12-s + (−3.07 + 3.07i)13-s + (1.64 − 3.50i)15-s + 4.48·16-s + (−0.850 + 0.850i)17-s + (−3.26 + 3.15i)18-s + 6.87i·19-s + ⋯ |
L(s) = 1 | + (0.755 − 0.755i)2-s + (−0.00867 + 0.999i)3-s − 0.142i·4-s + (−0.908 − 0.417i)5-s + (0.749 + 0.762i)6-s + (0.648 + 0.648i)8-s + (−0.999 − 0.0173i)9-s + (−1.00 + 0.371i)10-s + 0.275i·11-s + (0.142 + 0.00123i)12-s + (−0.854 + 0.854i)13-s + (0.425 − 0.905i)15-s + 1.12·16-s + (−0.206 + 0.206i)17-s + (−0.768 + 0.742i)18-s + 1.57i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.926911 + 1.03951i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.926911 + 1.03951i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.0150 - 1.73i)T \) |
| 5 | \( 1 + (2.03 + 0.933i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-1.06 + 1.06i)T - 2iT^{2} \) |
| 11 | \( 1 - 0.914iT - 11T^{2} \) |
| 13 | \( 1 + (3.07 - 3.07i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.850 - 0.850i)T - 17iT^{2} \) |
| 19 | \( 1 - 6.87iT - 19T^{2} \) |
| 23 | \( 1 + (1.38 + 1.38i)T + 23iT^{2} \) |
| 29 | \( 1 + 2.72T + 29T^{2} \) |
| 31 | \( 1 + 4.63T + 31T^{2} \) |
| 37 | \( 1 + (-0.567 - 0.567i)T + 37iT^{2} \) |
| 41 | \( 1 - 0.922iT - 41T^{2} \) |
| 43 | \( 1 + (4.80 - 4.80i)T - 43iT^{2} \) |
| 47 | \( 1 + (-7.41 + 7.41i)T - 47iT^{2} \) |
| 53 | \( 1 + (-7.79 - 7.79i)T + 53iT^{2} \) |
| 59 | \( 1 - 9.88T + 59T^{2} \) |
| 61 | \( 1 - 1.06T + 61T^{2} \) |
| 67 | \( 1 + (5.00 + 5.00i)T + 67iT^{2} \) |
| 71 | \( 1 + 0.557iT - 71T^{2} \) |
| 73 | \( 1 + (-1.54 + 1.54i)T - 73iT^{2} \) |
| 79 | \( 1 + 3.03iT - 79T^{2} \) |
| 83 | \( 1 + (2.38 + 2.38i)T + 83iT^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 + (-1.58 - 1.58i)T + 97iT^{2} \) |
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
−10.73909887732044977749051543281, −10.00590284741569136631729012150, −8.982874328646216143844070611780, −8.154430868349894386282126199715, −7.28251868579442506775208198708, −5.70532355852582721213642080732, −4.75155172241814218420370280912, −4.10140774147548991641175540805, −3.44534840805737572657549588554, −2.08562410349672752429386288934,
0.54624814076424060202431119551, 2.51861840194001763164343087153, 3.69424247682956588727284510657, 4.95492594384185352408054307311, 5.71749273754568652257494675299, 6.85646242583335950159441235605, 7.23395784258974416364438465560, 7.979923662146128399305248865639, 9.039775335418261430202877435166, 10.34225758157147326172034411141