L(s) = 1 | + 2-s + 4-s + 5-s + 2·7-s + 8-s + 10-s + 2·11-s + 2·13-s + 2·14-s + 16-s + 8·19-s + 20-s + 2·22-s + 8·23-s + 25-s + 2·26-s + 2·28-s + 2·29-s + 32-s + 2·35-s + 8·37-s + 8·38-s + 40-s + 8·41-s + 2·44-s + 8·46-s − 3·49-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 + 0.603·11-s + 0.554·13-s + 0.534·14-s + 1/4·16-s + 1.83·19-s + 0.223·20-s + 0.426·22-s + 1.66·23-s + 1/5·25-s + 0.392·26-s + 0.377·28-s + 0.371·29-s + 0.176·32-s + 0.338·35-s + 1.31·37-s + 1.29·38-s + 0.158·40-s + 1.24·41-s + 0.301·44-s + 1.17·46-s − 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.944465357\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.944465357\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 43 | \( 1 \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.37559672532389, −12.77659467253763, −12.38165807488662, −11.75577846828048, −11.38244648286837, −11.00640131199150, −10.63309022846019, −9.829223376641203, −9.364379428347868, −9.112097940089875, −8.342841305147348, −7.822018740181886, −7.406278372037063, −6.817336979569658, −6.342198231488121, −5.747037863886016, −5.372137017139842, −4.687913807553937, −4.500463758814736, −3.596586120632562, −3.170551657112456, −2.630547129307103, −1.875078412062388, −1.118248403965303, −0.9322502020280471,
0.9322502020280471, 1.118248403965303, 1.875078412062388, 2.630547129307103, 3.170551657112456, 3.596586120632562, 4.500463758814736, 4.687913807553937, 5.372137017139842, 5.747037863886016, 6.342198231488121, 6.817336979569658, 7.406278372037063, 7.822018740181886, 8.342841305147348, 9.112097940089875, 9.364379428347868, 9.829223376641203, 10.63309022846019, 11.00640131199150, 11.38244648286837, 11.75577846828048, 12.38165807488662, 12.77659467253763, 13.37559672532389