| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 2·7-s + 8-s + 9-s + 11-s − 12-s − 4·13-s + 2·14-s + 16-s + 6·17-s + 18-s + 4·19-s − 2·21-s + 22-s + 6·23-s − 24-s − 5·25-s − 4·26-s − 27-s + 2·28-s − 8·31-s + 32-s − 33-s + 6·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s − 1.10·13-s + 0.534·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.436·21-s + 0.213·22-s + 1.25·23-s − 0.204·24-s − 25-s − 0.784·26-s − 0.192·27-s + 0.377·28-s − 1.43·31-s + 0.176·32-s − 0.174·33-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55506 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55506 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.76072840984293, −14.22212588624248, −13.64518272326291, −13.10142758249823, −12.57044448212356, −12.00093017509604, −11.75249288077087, −11.24254834379269, −10.73638463880211, −10.05024274379191, −9.619487767986583, −9.142157184497605, −8.108697038770464, −7.795152873884944, −7.168349024209767, −6.863686966384258, −5.878914158924880, −5.525790670424890, −5.079368752526287, −4.548663303186098, −3.865103376818500, −3.187026409854268, −2.585339559261515, −1.583765024752358, −1.208608238018424, 0,
1.208608238018424, 1.583765024752358, 2.585339559261515, 3.187026409854268, 3.865103376818500, 4.548663303186098, 5.079368752526287, 5.525790670424890, 5.878914158924880, 6.863686966384258, 7.168349024209767, 7.795152873884944, 8.108697038770464, 9.142157184497605, 9.619487767986583, 10.05024274379191, 10.73638463880211, 11.24254834379269, 11.75249288077087, 12.00093017509604, 12.57044448212356, 13.10142758249823, 13.64518272326291, 14.22212588624248, 14.76072840984293
