| L(s) = 1 | + 2-s + 4-s + 5-s − 5·7-s + 8-s + 10-s + 11-s + 13-s − 5·14-s + 16-s + 19-s + 20-s + 22-s − 2·23-s + 25-s + 26-s − 5·28-s − 4·29-s − 31-s + 32-s − 5·35-s − 4·37-s + 38-s + 40-s + 6·41-s − 43-s + 44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.88·7-s + 0.353·8-s + 0.316·10-s + 0.301·11-s + 0.277·13-s − 1.33·14-s + 1/4·16-s + 0.229·19-s + 0.223·20-s + 0.213·22-s − 0.417·23-s + 1/5·25-s + 0.196·26-s − 0.944·28-s − 0.742·29-s − 0.179·31-s + 0.176·32-s − 0.845·35-s − 0.657·37-s + 0.162·38-s + 0.158·40-s + 0.937·41-s − 0.152·43-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 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 17 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−16.48338322123735, −15.87630329630174, −15.58051117945856, −14.82313901697631, −14.15601104545318, −13.69464373181739, −13.10602116723180, −12.74606152560037, −12.21966955391807, −11.54320212093265, −10.84245650544192, −10.11497309608252, −9.788534278589935, −9.086510750849476, −8.556432654856094, −7.376010670105161, −7.056270821763308, −6.263370405964127, −5.893008223526641, −5.317690471080550, −4.181574426552859, −3.739124828562825, −2.984046112486504, −2.380660958606211, −1.266304512420029, 0,
1.266304512420029, 2.380660958606211, 2.984046112486504, 3.739124828562825, 4.181574426552859, 5.317690471080550, 5.893008223526641, 6.263370405964127, 7.056270821763308, 7.376010670105161, 8.556432654856094, 9.086510750849476, 9.788534278589935, 10.11497309608252, 10.84245650544192, 11.54320212093265, 12.21966955391807, 12.74606152560037, 13.10602116723180, 13.69464373181739, 14.15601104545318, 14.82313901697631, 15.58051117945856, 15.87630329630174, 16.48338322123735
