| L(s) = 1 | − 3-s − 2·5-s − 2.82i·7-s + 9-s + 1.41i·11-s + 2.82i·13-s + 2·15-s − 2·17-s + (−1 − 4.24i)19-s + 2.82i·21-s + 1.41i·23-s − 25-s − 27-s + 7.07i·29-s + 6·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 1.06i·7-s + 0.333·9-s + 0.426i·11-s + 0.784i·13-s + 0.516·15-s − 0.485·17-s + (−0.229 − 0.973i)19-s + 0.617i·21-s + 0.294i·23-s − 0.200·25-s − 0.192·27-s + 1.31i·29-s + 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.327514 + 0.413685i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.327514 + 0.413685i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 + (1 + 4.24i)T \) |
| good | 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 2.82iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 23 | \( 1 - 1.41iT - 23T^{2} \) |
| 29 | \( 1 - 7.07iT - 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 11.3iT - 37T^{2} \) |
| 41 | \( 1 - 4.24iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 7.07iT - 47T^{2} \) |
| 53 | \( 1 - 9.89iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 12.7iT - 83T^{2} \) |
| 89 | \( 1 + 9.89iT - 89T^{2} \) |
| 97 | \( 1 + 2.82iT - 97T^{2} \) |
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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
−10.49009964094624038166094604380, −9.602276902858558845253128331582, −8.622150349541844026849169680908, −7.57309351257899498839697613529, −7.01432128788786399669100099115, −6.22263554687847045965021356679, −4.57377032110579241046194191784, −4.46532378233860291622697015290, −3.10144384723055400883782942775, −1.30409466606744166943503749909,
0.29126463668370005812740508918, 2.22504849284825007047214765418, 3.52518044091434973502996649912, 4.50440258527762922987341229427, 5.66044936410074556300768979170, 6.15289085109206836247261027225, 7.41683259987106922561770528688, 8.178868347312978205721783340220, 8.873051395106045766750002972322, 9.981165310540108862159012936753
