L(s) = 1 | − 2-s + 0.0361·3-s + 4-s + 5-s − 0.0361·6-s + 1.83·7-s − 8-s − 2.99·9-s − 10-s − 2.46·11-s + 0.0361·12-s + 2.39·13-s − 1.83·14-s + 0.0361·15-s + 16-s + 6.74·17-s + 2.99·18-s + 20-s + 0.0664·21-s + 2.46·22-s + 1.62·23-s − 0.0361·24-s + 25-s − 2.39·26-s − 0.216·27-s + 1.83·28-s + 3.30·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.0208·3-s + 0.5·4-s + 0.447·5-s − 0.0147·6-s + 0.694·7-s − 0.353·8-s − 0.999·9-s − 0.316·10-s − 0.743·11-s + 0.0104·12-s + 0.663·13-s − 0.491·14-s + 0.00933·15-s + 0.250·16-s + 1.63·17-s + 0.706·18-s + 0.223·20-s + 0.0144·21-s + 0.525·22-s + 0.337·23-s − 0.00737·24-s + 0.200·25-s − 0.469·26-s − 0.0417·27-s + 0.347·28-s + 0.613·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.509627595\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.509627595\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 - 0.0361T + 3T^{2} \) |
| 7 | \( 1 - 1.83T + 7T^{2} \) |
| 11 | \( 1 + 2.46T + 11T^{2} \) |
| 13 | \( 1 - 2.39T + 13T^{2} \) |
| 17 | \( 1 - 6.74T + 17T^{2} \) |
| 23 | \( 1 - 1.62T + 23T^{2} \) |
| 29 | \( 1 - 3.30T + 29T^{2} \) |
| 31 | \( 1 + 3.50T + 31T^{2} \) |
| 37 | \( 1 + 6.00T + 37T^{2} \) |
| 41 | \( 1 + 7.98T + 41T^{2} \) |
| 43 | \( 1 - 5.71T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 + 1.92T + 53T^{2} \) |
| 59 | \( 1 + 4.75T + 59T^{2} \) |
| 61 | \( 1 - 13.6T + 61T^{2} \) |
| 67 | \( 1 + 3.71T + 67T^{2} \) |
| 71 | \( 1 - 5.83T + 71T^{2} \) |
| 73 | \( 1 + 0.510T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 6.23T + 83T^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−8.474242620164301160125352759085, −8.006671039326122961179303294023, −7.27957022252101705319296414679, −6.30598873530587942455225348543, −5.50547664813474427803528201739, −5.11759905996759599550044746690, −3.64747910827765926271864631173, −2.85313337124490118650283456613, −1.88687895112688601019880438531, −0.814305962051955273069140660837,
0.814305962051955273069140660837, 1.88687895112688601019880438531, 2.85313337124490118650283456613, 3.64747910827765926271864631173, 5.11759905996759599550044746690, 5.50547664813474427803528201739, 6.30598873530587942455225348543, 7.27957022252101705319296414679, 8.006671039326122961179303294023, 8.474242620164301160125352759085