| 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 + 6·17-s + 8·19-s + 20-s − 22-s − 6·23-s + 25-s + 26-s + 2·28-s − 4·31-s + 32-s + 6·34-s + 2·35-s − 4·37-s + 8·38-s + 40-s + 6·41-s − 4·43-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 + 1.45·17-s + 1.83·19-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 − 0.718·31-s + 0.176·32-s + 1.02·34-s + 0.338·35-s − 0.657·37-s + 1.29·38-s + 0.158·40-s + 0.937·41-s − 0.609·43-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
| \(L(1)\) |
\(\approx\) |
\(4.909626230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.909626230\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 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
−16.04099689428366, −15.92003952355673, −14.93829582275170, −14.37813014502662, −14.15990016868232, −13.54555376616023, −12.96907772572702, −12.20659133563644, −11.85742429710201, −11.25482170452136, −10.62343781457391, −9.832679163025108, −9.645065355063815, −8.521495795710598, −7.974187130612298, −7.440316380671635, −6.780318355951358, −5.816439337491021, −5.423220575126072, −5.024382976522740, −3.944044138140186, −3.448738932354564, −2.565795136348015, −1.723451733819358, −0.9532179375307565,
0.9532179375307565, 1.723451733819358, 2.565795136348015, 3.448738932354564, 3.944044138140186, 5.024382976522740, 5.423220575126072, 5.816439337491021, 6.780318355951358, 7.440316380671635, 7.974187130612298, 8.521495795710598, 9.645065355063815, 9.832679163025108, 10.62343781457391, 11.25482170452136, 11.85742429710201, 12.20659133563644, 12.96907772572702, 13.54555376616023, 14.15990016868232, 14.37813014502662, 14.93829582275170, 15.92003952355673, 16.04099689428366
