L(s) = 1 | − 1.28i·2-s − i·3-s + 0.359·4-s + (2.05 − 0.876i)5-s − 1.28·6-s − i·7-s − 3.02i·8-s − 9-s + (−1.12 − 2.63i)10-s + 11-s − 0.359i·12-s − 4.33i·13-s − 1.28·14-s + (−0.876 − 2.05i)15-s − 3.15·16-s + 6.42i·17-s + ⋯ |
L(s) = 1 | − 0.905i·2-s − 0.577i·3-s + 0.179·4-s + (0.920 − 0.391i)5-s − 0.522·6-s − 0.377i·7-s − 1.06i·8-s − 0.333·9-s + (−0.354 − 0.833i)10-s + 0.301·11-s − 0.103i·12-s − 1.20i·13-s − 0.342·14-s + (−0.226 − 0.531i)15-s − 0.788·16-s + 1.55i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 + 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.125285971\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.125285971\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 + (-2.05 + 0.876i)T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 + 1.28iT - 2T^{2} \) |
| 13 | \( 1 + 4.33iT - 13T^{2} \) |
| 17 | \( 1 - 6.42iT - 17T^{2} \) |
| 19 | \( 1 + 2.73T + 19T^{2} \) |
| 23 | \( 1 + 3.91iT - 23T^{2} \) |
| 29 | \( 1 + 4.11T + 29T^{2} \) |
| 31 | \( 1 - 8.94T + 31T^{2} \) |
| 37 | \( 1 - 11.0iT - 37T^{2} \) |
| 41 | \( 1 + 3.74T + 41T^{2} \) |
| 43 | \( 1 - 10.5iT - 43T^{2} \) |
| 47 | \( 1 + 11.8iT - 47T^{2} \) |
| 53 | \( 1 + 8.18iT - 53T^{2} \) |
| 59 | \( 1 - 4.77T + 59T^{2} \) |
| 61 | \( 1 - 7.12T + 61T^{2} \) |
| 67 | \( 1 + 5.84iT - 67T^{2} \) |
| 71 | \( 1 + 0.841T + 71T^{2} \) |
| 73 | \( 1 + 0.456iT - 73T^{2} \) |
| 79 | \( 1 - 6.39T + 79T^{2} \) |
| 83 | \( 1 - 9.87iT - 83T^{2} \) |
| 89 | \( 1 + 8.67T + 89T^{2} \) |
| 97 | \( 1 - 2.64iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.876876636249059066896897063458, −8.538064029515617345596462252569, −8.040363185500184260203467608602, −6.54624678569972600356573729349, −6.39175928948011052112324292932, −5.13925605526166867749490133400, −3.91277377890053295919327606385, −2.82459587855640765935591044649, −1.85646011495796016919653965882, −0.926116144798390161167848741346,
1.97108856321362031002736671427, 2.84628019105511542843250963278, 4.31631900590688207994269160903, 5.33577067395640060226334428247, 5.94882734694534761571507988006, 6.82468373263804928789114098665, 7.37787488261746491031226228617, 8.670526645088723377366928528982, 9.242235013545378725273271331378, 9.915953305244479486878147944888