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
−9.440221256718840550841204307383, −8.608058198206992826255080156096, −8.004083576310768605506292214559, −7.11373760230580081571518093482, −6.57114764670937299145865532503, −5.19252909242407984762190193252, −4.37169567019189954286039121515, −3.63914112720517828083773685056, −2.70802627485331746646582743236, −1.29101926451057473091092356813,
0.834172367789482434231269979883, 1.96199282817752544882982398097, 3.18432987282353139826934682865, 4.20267284556659502241031055175, 5.05975148651272634143789810103, 5.85860587298693717787448299344, 7.03711109731225314218181592604, 7.81319296822253865529647039872, 8.274497096305548291138502841039, 9.200868645370788728467510859260