| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 2·7-s + 8-s + 9-s − 10-s + 2·11-s + 12-s + 13-s + 2·14-s − 15-s + 16-s + 18-s + 19-s − 20-s + 2·21-s + 2·22-s + 5·23-s + 24-s + 25-s + 26-s + 27-s + 2·28-s + 6·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s + 0.288·12-s + 0.277·13-s + 0.534·14-s − 0.258·15-s + 1/4·16-s + 0.235·18-s + 0.229·19-s − 0.223·20-s + 0.436·21-s + 0.426·22-s + 1.04·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.377·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.722997424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.722997424\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.20460279660650, −12.88615797018039, −12.37625653705266, −11.80669968170506, −11.43909205168698, −10.96651724033416, −10.61735310929838, −9.926561497868407, −9.264825029816604, −8.944212191244766, −8.422246730626585, −7.785366948797885, −7.478552770539676, −7.018345055754314, −6.292965035441416, −5.956528173803043, −5.140500665878181, −4.686633720682691, −4.344718972152892, −3.550343022810078, −3.351647645546441, −2.545104008573648, −2.013028723477324, −1.261509243541487, −0.7390471767154268,
0.7390471767154268, 1.261509243541487, 2.013028723477324, 2.545104008573648, 3.351647645546441, 3.550343022810078, 4.344718972152892, 4.686633720682691, 5.140500665878181, 5.956528173803043, 6.292965035441416, 7.018345055754314, 7.478552770539676, 7.785366948797885, 8.422246730626585, 8.944212191244766, 9.264825029816604, 9.926561497868407, 10.61735310929838, 10.96651724033416, 11.43909205168698, 11.80669968170506, 12.37625653705266, 12.88615797018039, 13.20460279660650
