| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s + 4·11-s + 12-s − 13-s + 15-s + 16-s − 6·17-s − 18-s + 20-s − 4·22-s + 8·23-s − 24-s + 25-s + 26-s + 27-s − 6·29-s − 30-s + 8·31-s − 32-s + 4·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s + 0.288·12-s − 0.277·13-s + 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.223·20-s − 0.852·22-s + 1.66·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 1.11·29-s − 0.182·30-s + 1.43·31-s − 0.176·32-s + 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.250496419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.250496419\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−13.35135391611381, −12.96303078338819, −12.57561138093372, −11.78108354402824, −11.41569250297205, −10.97632787076618, −10.53795548812907, −9.744759533526017, −9.462258115777113, −9.132194159270668, −8.659094468896426, −8.207086594916941, −7.474534639330588, −7.097582989600523, −6.578477410885193, −6.175863042159101, −5.561344836130153, −4.655144223503600, −4.386221008756392, −3.675918318611472, −2.905681293037062, −2.496740328407528, −1.891063593912267, −1.174137675707518, −0.6316092594028959,
0.6316092594028959, 1.174137675707518, 1.891063593912267, 2.496740328407528, 2.905681293037062, 3.675918318611472, 4.386221008756392, 4.655144223503600, 5.561344836130153, 6.175863042159101, 6.578477410885193, 7.097582989600523, 7.474534639330588, 8.207086594916941, 8.659094468896426, 9.132194159270668, 9.462258115777113, 9.744759533526017, 10.53795548812907, 10.97632787076618, 11.41569250297205, 11.78108354402824, 12.57561138093372, 12.96303078338819, 13.35135391611381
