| L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s + 2·11-s + 4·13-s + 14-s + 16-s + 6·17-s − 4·19-s + 20-s + 2·22-s + 23-s + 25-s + 4·26-s + 28-s − 2·29-s + 2·31-s + 32-s + 6·34-s + 35-s + 4·37-s − 4·38-s + 40-s + 2·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s + 0.603·11-s + 1.10·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.223·20-s + 0.426·22-s + 0.208·23-s + 1/5·25-s + 0.784·26-s + 0.188·28-s − 0.371·29-s + 0.359·31-s + 0.176·32-s + 1.02·34-s + 0.169·35-s + 0.657·37-s − 0.648·38-s + 0.158·40-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.184653161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.184653161\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−16.10572591336038, −15.40215394157608, −14.83957560278335, −14.45341911549985, −13.84163211930038, −13.41520765201174, −12.78923104079939, −12.18492344269480, −11.73156653509228, −10.97626110883613, −10.59979192373567, −9.927127822250711, −9.157005093558265, −8.635188690998541, −7.893057903630730, −7.303195535319660, −6.514173269305122, −5.881979732857415, −5.606204643669510, −4.595116118285024, −4.067837996348545, −3.351936000096759, −2.572072751679161, −1.625653913701199, −0.9824850512271249,
0.9824850512271249, 1.625653913701199, 2.572072751679161, 3.351936000096759, 4.067837996348545, 4.595116118285024, 5.606204643669510, 5.881979732857415, 6.514173269305122, 7.303195535319660, 7.893057903630730, 8.635188690998541, 9.157005093558265, 9.927127822250711, 10.59979192373567, 10.97626110883613, 11.73156653509228, 12.18492344269480, 12.78923104079939, 13.41520765201174, 13.84163211930038, 14.45341911549985, 14.83957560278335, 15.40215394157608, 16.10572591336038
