| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 7-s − 8-s + 9-s − 10-s − 3·11-s − 12-s + 6·13-s − 14-s − 15-s + 16-s + 3·17-s − 18-s + 8·19-s + 20-s − 21-s + 3·22-s + 23-s + 24-s + 25-s − 6·26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.904·11-s − 0.288·12-s + 1.66·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 1.83·19-s + 0.223·20-s − 0.218·21-s + 0.639·22-s + 0.208·23-s + 0.204·24-s + 1/5·25-s − 1.17·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.824155529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.824155529\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−12.73168084102047, −12.16220460034267, −11.59746295246164, −11.36354835463579, −10.80552538599212, −10.39307802092564, −10.14842329552812, −9.478804963500183, −9.038468569619980, −8.669715080043984, −7.963420925051209, −7.648565357644792, −7.279911688419572, −6.572975313884780, −6.066880168364169, −5.691235422827571, −5.242766346779949, −4.819211337727495, −3.943672361371438, −3.443740074355679, −2.894388201874026, −2.254256593132291, −1.519397854932375, −0.9852594789426817, −0.6569651873618610,
0.6569651873618610, 0.9852594789426817, 1.519397854932375, 2.254256593132291, 2.894388201874026, 3.443740074355679, 3.943672361371438, 4.819211337727495, 5.242766346779949, 5.691235422827571, 6.066880168364169, 6.572975313884780, 7.279911688419572, 7.648565357644792, 7.963420925051209, 8.669715080043984, 9.038468569619980, 9.478804963500183, 10.14842329552812, 10.39307802092564, 10.80552538599212, 11.36354835463579, 11.59746295246164, 12.16220460034267, 12.73168084102047
