| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s + 6·11-s + 12-s + 13-s − 14-s + 16-s − 17-s + 18-s + 19-s − 21-s + 6·22-s + 4·23-s + 24-s + 26-s + 27-s − 28-s − 8·29-s − 3·31-s + 32-s + 6·33-s − 34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 1.80·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.229·19-s − 0.218·21-s + 1.27·22-s + 0.834·23-s + 0.204·24-s + 0.196·26-s + 0.192·27-s − 0.188·28-s − 1.48·29-s − 0.538·31-s + 0.176·32-s + 1.04·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.971435955\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.971435955\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 11 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−12.82964258368552, −12.69344600111833, −12.03633028360681, −11.57843487473608, −11.09264536562131, −10.87772557241697, −9.981156298997337, −9.588098659949737, −9.210815629586763, −8.766527936661025, −8.296875290918068, −7.494894034275454, −7.180535955853881, −6.760966501651632, −6.137669918996402, −5.859481964801474, −5.097490480007127, −4.589354532777434, −3.907225794761649, −3.634993512615586, −3.293282884032851, −2.337881116903398, −2.048973568091007, −1.203156180562330, −0.7102810388348740,
0.7102810388348740, 1.203156180562330, 2.048973568091007, 2.337881116903398, 3.293282884032851, 3.634993512615586, 3.907225794761649, 4.589354532777434, 5.097490480007127, 5.859481964801474, 6.137669918996402, 6.760966501651632, 7.180535955853881, 7.494894034275454, 8.296875290918068, 8.766527936661025, 9.210815629586763, 9.588098659949737, 9.981156298997337, 10.87772557241697, 11.09264536562131, 11.57843487473608, 12.03633028360681, 12.69344600111833, 12.82964258368552
