| 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 + 2·23-s + 25-s + 26-s + 2·28-s + 4·31-s + 32-s − 2·34-s + 2·35-s + 4·37-s + 40-s + 6·41-s − 4·43-s − 44-s + 2·46-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 + 0.417·23-s + 1/5·25-s + 0.196·26-s + 0.377·28-s + 0.718·31-s + 0.176·32-s − 0.342·34-s + 0.338·35-s + 0.657·37-s + 0.158·40-s + 0.937·41-s − 0.609·43-s − 0.150·44-s + 0.294·46-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
| \(L(1)\) |
\(\approx\) |
\(4.479055367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.479055367\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−16.15693676686292, −15.62664436033546, −14.95864537812379, −14.69522453732943, −13.84143997588739, −13.57180789932296, −13.02691242331637, −12.27973296869321, −11.84039856031359, −11.05772261806494, −10.74638048289942, −10.09972899972078, −9.228453875493320, −8.777247061334284, −7.845088198020603, −7.550645396003849, −6.538671924580890, −6.151351864374017, −5.356835739874562, −4.772347036600872, −4.236006552614773, −3.299882031034482, −2.528676654296850, −1.835444265237266, −0.8745144356828046,
0.8745144356828046, 1.835444265237266, 2.528676654296850, 3.299882031034482, 4.236006552614773, 4.772347036600872, 5.356835739874562, 6.151351864374017, 6.538671924580890, 7.550645396003849, 7.845088198020603, 8.777247061334284, 9.228453875493320, 10.09972899972078, 10.74638048289942, 11.05772261806494, 11.84039856031359, 12.27973296869321, 13.02691242331637, 13.57180789932296, 13.84143997588739, 14.69522453732943, 14.95864537812379, 15.62664436033546, 16.15693676686292
