| L(s) = 1 | + (−1.71 − 0.265i)3-s − 2.00i·5-s + 1.70i·7-s + (2.85 + 0.907i)9-s − 1.65·11-s − 13-s + (−0.532 + 3.43i)15-s + 1.01i·17-s + 2.82i·19-s + (0.451 − 2.91i)21-s − 0.858·23-s + 0.963·25-s + (−4.65 − 2.31i)27-s − 1.49i·29-s + 6.85i·31-s + ⋯ |
| L(s) = 1 | + (−0.988 − 0.153i)3-s − 0.898i·5-s + 0.644i·7-s + (0.953 + 0.302i)9-s − 0.500·11-s − 0.277·13-s + (−0.137 + 0.887i)15-s + 0.245i·17-s + 0.648i·19-s + (0.0985 − 0.636i)21-s − 0.179·23-s + 0.192·25-s + (−0.895 − 0.444i)27-s − 0.278i·29-s + 1.23i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.590i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.806 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9503204038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9503204038\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.71 + 0.265i)T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2.00iT - 5T^{2} \) |
| 7 | \( 1 - 1.70iT - 7T^{2} \) |
| 11 | \( 1 + 1.65T + 11T^{2} \) |
| 17 | \( 1 - 1.01iT - 17T^{2} \) |
| 19 | \( 1 - 2.82iT - 19T^{2} \) |
| 23 | \( 1 + 0.858T + 23T^{2} \) |
| 29 | \( 1 + 1.49iT - 29T^{2} \) |
| 31 | \( 1 - 6.85iT - 31T^{2} \) |
| 37 | \( 1 + 1.09T + 37T^{2} \) |
| 41 | \( 1 - 1.83iT - 41T^{2} \) |
| 43 | \( 1 - 1.74iT - 43T^{2} \) |
| 47 | \( 1 - 5.06T + 47T^{2} \) |
| 53 | \( 1 + 1.18iT - 53T^{2} \) |
| 59 | \( 1 - 9.69T + 59T^{2} \) |
| 61 | \( 1 - 9.75T + 61T^{2} \) |
| 67 | \( 1 - 9.64iT - 67T^{2} \) |
| 71 | \( 1 - 0.904T + 71T^{2} \) |
| 73 | \( 1 + 0.939T + 73T^{2} \) |
| 79 | \( 1 - 4.69iT - 79T^{2} \) |
| 83 | \( 1 - 8.81T + 83T^{2} \) |
| 89 | \( 1 - 16.0iT - 89T^{2} \) |
| 97 | \( 1 - 8.30T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.921207538424017207363033926100, −8.913998311577981461501875686560, −8.214024312537455018117041615829, −7.26966843792437619233940746374, −6.31106321401873725102554194064, −5.42677239364995495822912086253, −4.98025372627671193224156178839, −3.89471224649995174161360336076, −2.30968847536684182906086353923, −1.03864429434169390297242258450,
0.56445400834702101387588147220, 2.31391066934688076480648271774, 3.56123020579965936509625880080, 4.53616574447135918849708745340, 5.42700387434470006668150538387, 6.35340875626620843927309106686, 7.12450217412268333531045976827, 7.60806773885336020050212914428, 8.947048106355009265542885071194, 9.982067923339402088143165507450
