L(s) = 1 | + (0.707 − 0.707i)3-s + (−1.50 − 1.65i)5-s + (2.20 + 1.46i)7-s − 1.00i·9-s + 1.46·11-s + (0.887 − 0.887i)13-s + (−2.23 − 0.103i)15-s + (−2.10 − 2.10i)17-s + 3.95·19-s + (2.59 − 0.526i)21-s + (4.13 + 4.13i)23-s + (−0.462 + 4.97i)25-s + (−0.707 − 0.707i)27-s − 5.18i·29-s − 6.10i·31-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (−0.673 − 0.739i)5-s + (0.833 + 0.552i)7-s − 0.333i·9-s + 0.441·11-s + (0.246 − 0.246i)13-s + (−0.576 − 0.0267i)15-s + (−0.510 − 0.510i)17-s + 0.908·19-s + (0.565 − 0.114i)21-s + (0.861 + 0.861i)23-s + (−0.0925 + 0.995i)25-s + (−0.136 − 0.136i)27-s − 0.962i·29-s − 1.09i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.975030113\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.975030113\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (1.50 + 1.65i)T \) |
| 7 | \( 1 + (-2.20 - 1.46i)T \) |
good | 11 | \( 1 - 1.46T + 11T^{2} \) |
| 13 | \( 1 + (-0.887 + 0.887i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.10 + 2.10i)T + 17iT^{2} \) |
| 19 | \( 1 - 3.95T + 19T^{2} \) |
| 23 | \( 1 + (-4.13 - 4.13i)T + 23iT^{2} \) |
| 29 | \( 1 + 5.18iT - 29T^{2} \) |
| 31 | \( 1 + 6.10iT - 31T^{2} \) |
| 37 | \( 1 + (-2.25 + 2.25i)T - 37iT^{2} \) |
| 41 | \( 1 + 0.769iT - 41T^{2} \) |
| 43 | \( 1 + (-5.18 - 5.18i)T + 43iT^{2} \) |
| 47 | \( 1 + (8.57 + 8.57i)T + 47iT^{2} \) |
| 53 | \( 1 + (0.544 + 0.544i)T + 53iT^{2} \) |
| 59 | \( 1 - 3.19T + 59T^{2} \) |
| 61 | \( 1 + 1.42iT - 61T^{2} \) |
| 67 | \( 1 + (-5.93 + 5.93i)T - 67iT^{2} \) |
| 71 | \( 1 + 7.62T + 71T^{2} \) |
| 73 | \( 1 + (-6.81 + 6.81i)T - 73iT^{2} \) |
| 79 | \( 1 - 4.52iT - 79T^{2} \) |
| 83 | \( 1 + (-6.75 + 6.75i)T - 83iT^{2} \) |
| 89 | \( 1 - 1.19T + 89T^{2} \) |
| 97 | \( 1 + (8.68 + 8.68i)T + 97iT^{2} \) |
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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
−9.200868645370788728467510859260, −8.274497096305548291138502841039, −7.81319296822253865529647039872, −7.03711109731225314218181592604, −5.85860587298693717787448299344, −5.05975148651272634143789810103, −4.20267284556659502241031055175, −3.18432987282353139826934682865, −1.96199282817752544882982398097, −0.834172367789482434231269979883,
1.29101926451057473091092356813, 2.70802627485331746646582743236, 3.63914112720517828083773685056, 4.37169567019189954286039121515, 5.19252909242407984762190193252, 6.57114764670937299145865532503, 7.11373760230580081571518093482, 8.004083576310768605506292214559, 8.608058198206992826255080156096, 9.440221256718840550841204307383