L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s − 2·11-s + 4·13-s + 16-s − 6·17-s − 19-s − 20-s + 2·22-s + 8·23-s + 25-s − 4·26-s − 6·29-s + 8·31-s − 32-s + 6·34-s − 8·37-s + 38-s + 40-s + 12·41-s − 2·44-s − 8·46-s − 50-s + 4·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 0.603·11-s + 1.10·13-s + 1/4·16-s − 1.45·17-s − 0.229·19-s − 0.223·20-s + 0.426·22-s + 1.66·23-s + 1/5·25-s − 0.784·26-s − 1.11·29-s + 1.43·31-s − 0.176·32-s + 1.02·34-s − 1.31·37-s + 0.162·38-s + 0.158·40-s + 1.87·41-s − 0.301·44-s − 1.17·46-s − 0.141·50-s + 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.671646154\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.671646154\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 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
−13.89915086994016, −13.24338315611149, −13.06337826462109, −12.50399891995120, −11.75520985084871, −11.31765452981974, −10.86823485859966, −10.67570022895573, −9.949350317284167, −9.261183760897174, −8.891163083674697, −8.448543413471179, −7.999087235134338, −7.340379728498771, −6.800527481129215, −6.493991760427445, −5.705074609589603, −5.155396769984987, −4.507183918252555, −3.817004484701336, −3.333340021617982, −2.439795758841632, −2.122189310207560, −1.003110174847582, −0.5690617450194681,
0.5690617450194681, 1.003110174847582, 2.122189310207560, 2.439795758841632, 3.333340021617982, 3.817004484701336, 4.507183918252555, 5.155396769984987, 5.705074609589603, 6.493991760427445, 6.800527481129215, 7.340379728498771, 7.999087235134338, 8.448543413471179, 8.891163083674697, 9.261183760897174, 9.949350317284167, 10.67570022895573, 10.86823485859966, 11.31765452981974, 11.75520985084871, 12.50399891995120, 13.06337826462109, 13.24338315611149, 13.89915086994016