L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.279 + 1.70i)3-s + (−0.866 + 0.499i)4-s + (−2.00 + 0.991i)5-s + (−1.72 + 0.172i)6-s + (−2.64 − 0.0981i)7-s + (−0.707 − 0.707i)8-s + (−2.84 − 0.956i)9-s + (−1.47 − 1.67i)10-s + (0.989 − 1.71i)11-s + (−0.612 − 1.62i)12-s + (3.22 + 0.864i)13-s + (−0.589 − 2.57i)14-s + (−1.13 − 3.70i)15-s + (0.500 − 0.866i)16-s + (−0.750 − 0.750i)17-s + ⋯ |
L(s) = 1 | + (0.183 + 0.683i)2-s + (−0.161 + 0.986i)3-s + (−0.433 + 0.249i)4-s + (−0.896 + 0.443i)5-s + (−0.703 + 0.0703i)6-s + (−0.999 − 0.0370i)7-s + (−0.249 − 0.249i)8-s + (−0.947 − 0.318i)9-s + (−0.466 − 0.530i)10-s + (0.298 − 0.516i)11-s + (−0.176 − 0.467i)12-s + (0.895 + 0.239i)13-s + (−0.157 − 0.689i)14-s + (−0.292 − 0.956i)15-s + (0.125 − 0.216i)16-s + (−0.182 − 0.182i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.154 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.154 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0826690 - 0.0707298i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0826690 - 0.0707298i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 - 0.965i)T \) |
| 3 | \( 1 + (0.279 - 1.70i)T \) |
| 5 | \( 1 + (2.00 - 0.991i)T \) |
| 7 | \( 1 + (2.64 + 0.0981i)T \) |
good | 11 | \( 1 + (-0.989 + 1.71i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.22 - 0.864i)T + (11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + (0.750 + 0.750i)T + 17iT^{2} \) |
| 19 | \( 1 + 2.29T + 19T^{2} \) |
| 23 | \( 1 + (0.726 - 2.70i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (5.55 + 3.20i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.55 + 2.05i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.26 + 1.26i)T - 37iT^{2} \) |
| 41 | \( 1 + (10.4 - 6.00i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.95 - 1.59i)T + (37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (10.3 - 2.77i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-6.80 - 6.80i)T + 53iT^{2} \) |
| 59 | \( 1 + (5.75 + 9.97i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (8.62 + 4.97i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.36 + 0.632i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 1.00T + 71T^{2} \) |
| 73 | \( 1 + (-4.70 + 4.70i)T - 73iT^{2} \) |
| 79 | \( 1 + (10.3 + 5.99i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.404 + 1.50i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + 3.56T + 89T^{2} \) |
| 97 | \( 1 + (-13.3 + 3.57i)T + (84.0 - 48.5i)T^{2} \) |
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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
−11.30642596562802791531314127515, −10.33645662439777357547564109303, −9.458223407298725612895918668398, −8.660090542826822485430974341292, −7.80865339717291148104072296960, −6.51208824233025628248433256767, −6.10451258567819982996779725425, −4.73382989906145861842159180297, −3.75484432186381883368445001928, −3.19961447015344225394407446979,
0.05771653370295077689978772625, 1.57080124336921826646729607335, 3.07023865636263297118753489092, 4.00198040400324697707337891445, 5.27997005787896871896834011920, 6.40998528132427969160931381956, 7.14346373413823518276396004840, 8.389908248485601557805990246217, 8.837515027347822051018761537942, 10.10251365254162301225978957170