L(s) = 1 | + 2-s + (−1.63 − 0.571i)3-s + 4-s + (−1.46 − 1.69i)5-s + (−1.63 − 0.571i)6-s + 2.83i·7-s + 8-s + (2.34 + 1.86i)9-s + (−1.46 − 1.69i)10-s − 3.85·11-s + (−1.63 − 0.571i)12-s + 2.42·13-s + 2.83i·14-s + (1.42 + 3.60i)15-s + 16-s + 4.97i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.943 − 0.330i)3-s + 0.5·4-s + (−0.654 − 0.756i)5-s + (−0.667 − 0.233i)6-s + 1.07i·7-s + 0.353·8-s + (0.782 + 0.623i)9-s + (−0.462 − 0.534i)10-s − 1.16·11-s + (−0.471 − 0.165i)12-s + 0.672·13-s + 0.758i·14-s + (0.368 + 0.929i)15-s + 0.250·16-s + 1.20i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.394i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.919 - 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.49921 + 0.307911i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.49921 + 0.307911i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (1.63 + 0.571i)T \) |
| 5 | \( 1 + (1.46 + 1.69i)T \) |
| 31 | \( 1 + (0.156 + 5.56i)T \) |
good | 7 | \( 1 - 2.83iT - 7T^{2} \) |
| 11 | \( 1 + 3.85T + 11T^{2} \) |
| 13 | \( 1 - 2.42T + 13T^{2} \) |
| 17 | \( 1 - 4.97iT - 17T^{2} \) |
| 19 | \( 1 - 8.42T + 19T^{2} \) |
| 23 | \( 1 + 0.144iT - 23T^{2} \) |
| 29 | \( 1 - 4.43T + 29T^{2} \) |
| 37 | \( 1 - 8.07T + 37T^{2} \) |
| 41 | \( 1 - 7.56iT - 41T^{2} \) |
| 43 | \( 1 + 0.244T + 43T^{2} \) |
| 47 | \( 1 + 0.441T + 47T^{2} \) |
| 53 | \( 1 - 9.87iT - 53T^{2} \) |
| 59 | \( 1 + 11.9iT - 59T^{2} \) |
| 61 | \( 1 - 10.1iT - 61T^{2} \) |
| 67 | \( 1 - 1.92iT - 67T^{2} \) |
| 71 | \( 1 + 0.737iT - 71T^{2} \) |
| 73 | \( 1 - 4.18T + 73T^{2} \) |
| 79 | \( 1 - 14.3iT - 79T^{2} \) |
| 83 | \( 1 + 4.53iT - 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 + 11.9iT - 97T^{2} \) |
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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
−10.36413933299334083366278986444, −9.323019096699039800943270278251, −8.066967347803081732283027507135, −7.72742475742706739877368713977, −6.35742796457272355713578782801, −5.61109978792190253129869266361, −5.07663116438756230854874750264, −4.05548600622243448408694533551, −2.70880859583094959604758114574, −1.20646370397255639317080711746,
0.77701163969740150476489603749, 2.95093841101689908543404747747, 3.74838243357690579399117549477, 4.76904673329274829290488391667, 5.47046638766297227210180925376, 6.63259223463835226833315692397, 7.26470206511300206989472730585, 7.88743052930214471717474215602, 9.538275447705467799685522768916, 10.44937428711776657184738609893