L(s) = 1 | + (0.707 + 0.707i)2-s + (0.867 − 1.49i)3-s + 1.00i·4-s + (0.850 − 2.06i)5-s + (1.67 − 0.446i)6-s + (0.811 − 0.811i)7-s + (−0.707 + 0.707i)8-s + (−1.49 − 2.60i)9-s + (2.06 − 0.861i)10-s + 1.25i·11-s + (1.49 + 0.867i)12-s + (3.40 + 3.40i)13-s + 1.14·14-s + (−2.36 − 3.06i)15-s − 1.00·16-s + (−3.04 − 3.04i)17-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (0.500 − 0.865i)3-s + 0.500i·4-s + (0.380 − 0.924i)5-s + (0.683 − 0.182i)6-s + (0.306 − 0.306i)7-s + (−0.250 + 0.250i)8-s + (−0.498 − 0.867i)9-s + (0.652 − 0.272i)10-s + 0.377i·11-s + (0.432 + 0.250i)12-s + (0.944 + 0.944i)13-s + 0.306·14-s + (−0.610 − 0.792i)15-s − 0.250·16-s + (−0.737 − 0.737i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.23940 - 0.795803i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.23940 - 0.795803i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (-0.867 + 1.49i)T \) |
| 5 | \( 1 + (-0.850 + 2.06i)T \) |
| 19 | \( 1 + iT \) |
good | 7 | \( 1 + (-0.811 + 0.811i)T - 7iT^{2} \) |
| 11 | \( 1 - 1.25iT - 11T^{2} \) |
| 13 | \( 1 + (-3.40 - 3.40i)T + 13iT^{2} \) |
| 17 | \( 1 + (3.04 + 3.04i)T + 17iT^{2} \) |
| 23 | \( 1 + (-5.51 + 5.51i)T - 23iT^{2} \) |
| 29 | \( 1 - 5.49T + 29T^{2} \) |
| 31 | \( 1 + 5.80T + 31T^{2} \) |
| 37 | \( 1 + (7.08 - 7.08i)T - 37iT^{2} \) |
| 41 | \( 1 + 0.259iT - 41T^{2} \) |
| 43 | \( 1 + (-5.60 - 5.60i)T + 43iT^{2} \) |
| 47 | \( 1 + (-6.12 - 6.12i)T + 47iT^{2} \) |
| 53 | \( 1 + (0.465 - 0.465i)T - 53iT^{2} \) |
| 59 | \( 1 + 9.08T + 59T^{2} \) |
| 61 | \( 1 - 6.32T + 61T^{2} \) |
| 67 | \( 1 + (2.40 - 2.40i)T - 67iT^{2} \) |
| 71 | \( 1 + 0.747iT - 71T^{2} \) |
| 73 | \( 1 + (-5.45 - 5.45i)T + 73iT^{2} \) |
| 79 | \( 1 + 4.84iT - 79T^{2} \) |
| 83 | \( 1 + (4.22 - 4.22i)T - 83iT^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + (12.5 - 12.5i)T - 97iT^{2} \) |
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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
−10.83313439806688216912644449852, −9.213794685713038139365398418861, −8.870343476006355280950216626016, −7.936877319512891787286359942112, −6.90303697786915612312697854360, −6.30667942306118108792359020920, −5.00025016655674006944153648587, −4.17330457316052785556349225736, −2.63204954270154254421751256607, −1.26617833647278509302043991611,
2.01290735436846636147843742689, 3.19901920914129738681001608528, 3.82769275145354118365314645107, 5.29131839239355801780258972870, 5.89465704343369640021846760534, 7.22224112380885562474202716933, 8.475764070074400662945877567347, 9.156353274242218182182889278439, 10.27656020046323854732114042219, 10.82608619097721990582132431870