| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 4·7-s − 8-s + 9-s − 10-s + 4·11-s + 12-s + 13-s + 4·14-s + 15-s + 16-s + 2·17-s − 18-s + 20-s − 4·21-s − 4·22-s + 4·23-s − 24-s + 25-s − 26-s + 27-s − 4·28-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s + 0.288·12-s + 0.277·13-s + 1.06·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.223·20-s − 0.872·21-s − 0.852·22-s + 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.755·28-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.149185112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.149185112\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.84553327858890, −12.14167403631781, −11.69280799309404, −11.05751951208302, −10.81939751854507, −10.02326915251876, −9.726010387578022, −9.491148798951124, −8.943824770520645, −8.770419331924342, −8.061660523749179, −7.455505587582886, −7.052538984974219, −6.665514753462028, −6.157630416300274, −5.779775121381421, −5.216685578200456, −4.258203506063068, −3.881381981326448, −3.368761190721891, −2.867717351924257, −2.361879797067125, −1.711338234366580, −0.9736025040462348, −0.5998711734440186,
0.5998711734440186, 0.9736025040462348, 1.711338234366580, 2.361879797067125, 2.867717351924257, 3.368761190721891, 3.881381981326448, 4.258203506063068, 5.216685578200456, 5.779775121381421, 6.157630416300274, 6.665514753462028, 7.052538984974219, 7.455505587582886, 8.061660523749179, 8.770419331924342, 8.943824770520645, 9.491148798951124, 9.726010387578022, 10.02326915251876, 10.81939751854507, 11.05751951208302, 11.69280799309404, 12.14167403631781, 12.84553327858890
