| L(s) = 1 | + 2·5-s − 2.82i·7-s − 1.41i·11-s + 2.82i·13-s + 2·17-s + (−1 − 4.24i)19-s − 1.41i·23-s − 25-s − 7.07i·29-s + 6·31-s − 5.65i·35-s + 11.3i·37-s − 4.24i·41-s − 7.07i·47-s − 1.00·49-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.06i·7-s − 0.426i·11-s + 0.784i·13-s + 0.485·17-s + (−0.229 − 0.973i)19-s − 0.294i·23-s − 0.200·25-s − 1.31i·29-s + 1.07·31-s − 0.956i·35-s + 1.85i·37-s − 0.662i·41-s − 1.03i·47-s − 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.999544742\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.999544742\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 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
−8.628418519873205653505689124771, −7.979024601144416548478720392696, −6.93046491031248673035674699290, −6.51730782226577339672886193519, −5.62177203256154037941756271806, −4.68256906960174965616361121885, −3.96150018981954052938822647874, −2.85744984246124962714683867956, −1.83543257837750060927044170708, −0.66004913437722848509469054681,
1.36328994955260503026483027157, 2.32802530422637219936668147700, 3.13231266049494287666858825674, 4.30052410883299608639780508916, 5.53216175885071598134879869818, 5.62804488175347957839789838602, 6.54422947909739461180924384449, 7.57928460845954272202522472185, 8.247950402571882068434279879202, 9.113230658535619010469194486066
