| L(s) = 1 | + 2-s + 4-s − 5-s − 2·7-s + 8-s − 10-s + 11-s − 13-s − 2·14-s + 16-s − 2·17-s − 20-s + 22-s + 6·23-s + 25-s − 26-s − 2·28-s + 4·29-s − 4·31-s + 32-s − 2·34-s + 2·35-s − 8·37-s − 40-s + 6·41-s + 4·43-s + 44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s − 0.277·13-s − 0.534·14-s + 1/4·16-s − 0.485·17-s − 0.223·20-s + 0.213·22-s + 1.25·23-s + 1/5·25-s − 0.196·26-s − 0.377·28-s + 0.742·29-s − 0.718·31-s + 0.176·32-s − 0.342·34-s + 0.338·35-s − 1.31·37-s − 0.158·40-s + 0.937·41-s + 0.609·43-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−16.51371388289448, −15.83467184774634, −15.44073912177542, −14.90489039319176, −14.23467625062439, −13.81879659470158, −12.99432898178182, −12.66154566029587, −12.23274633111406, −11.27667786453211, −11.16303751639295, −10.28913092767881, −9.671800770238019, −9.006309298153612, −8.411712158744534, −7.566545818726678, −6.936881788322264, −6.559386185357597, −5.778966782002852, −4.996943210269023, −4.465659560902552, −3.601928609104749, −3.128488175236162, −2.313330205928469, −1.232374431763789, 0,
1.232374431763789, 2.313330205928469, 3.128488175236162, 3.601928609104749, 4.465659560902552, 4.996943210269023, 5.778966782002852, 6.559386185357597, 6.936881788322264, 7.566545818726678, 8.411712158744534, 9.006309298153612, 9.671800770238019, 10.28913092767881, 11.16303751639295, 11.27667786453211, 12.23274633111406, 12.66154566029587, 12.99432898178182, 13.81879659470158, 14.23467625062439, 14.90489039319176, 15.44073912177542, 15.83467184774634, 16.51371388289448
