| L(s) = 1 | + 2-s + 1.28·3-s + 4-s + 3.69·5-s + 1.28·6-s − 0.442·7-s + 8-s − 1.35·9-s + 3.69·10-s − 4.02·11-s + 1.28·12-s + 4.89·13-s − 0.442·14-s + 4.74·15-s + 16-s + 0.266·17-s − 1.35·18-s + 3.69·20-s − 0.568·21-s − 4.02·22-s − 9.20·23-s + 1.28·24-s + 8.65·25-s + 4.89·26-s − 5.58·27-s − 0.442·28-s − 0.223·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.741·3-s + 0.5·4-s + 1.65·5-s + 0.524·6-s − 0.167·7-s + 0.353·8-s − 0.450·9-s + 1.16·10-s − 1.21·11-s + 0.370·12-s + 1.35·13-s − 0.118·14-s + 1.22·15-s + 0.250·16-s + 0.0647·17-s − 0.318·18-s + 0.826·20-s − 0.123·21-s − 0.859·22-s − 1.92·23-s + 0.262·24-s + 1.73·25-s + 0.959·26-s − 1.07·27-s − 0.0836·28-s − 0.0415·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.454011826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.454011826\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 1.28T + 3T^{2} \) |
| 5 | \( 1 - 3.69T + 5T^{2} \) |
| 7 | \( 1 + 0.442T + 7T^{2} \) |
| 11 | \( 1 + 4.02T + 11T^{2} \) |
| 13 | \( 1 - 4.89T + 13T^{2} \) |
| 17 | \( 1 - 0.266T + 17T^{2} \) |
| 23 | \( 1 + 9.20T + 23T^{2} \) |
| 29 | \( 1 + 0.223T + 29T^{2} \) |
| 31 | \( 1 - 3.47T + 31T^{2} \) |
| 37 | \( 1 - 1.44T + 37T^{2} \) |
| 41 | \( 1 + 7.86T + 41T^{2} \) |
| 43 | \( 1 + 5.63T + 43T^{2} \) |
| 47 | \( 1 - 2.19T + 47T^{2} \) |
| 53 | \( 1 - 9.94T + 53T^{2} \) |
| 59 | \( 1 + 3.50T + 59T^{2} \) |
| 61 | \( 1 - 4.07T + 61T^{2} \) |
| 67 | \( 1 - 0.147T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 - 1.42T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + 3.28T + 83T^{2} \) |
| 89 | \( 1 + 5.96T + 89T^{2} \) |
| 97 | \( 1 - 4.08T + 97T^{2} \) |
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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
−10.26781599779401054651658692830, −9.724681875952850212372395349151, −8.590644629105691064086774606872, −7.993353682382915666008091576358, −6.53349138395219407464163964289, −5.87828291361757467118536060459, −5.19489747629461411283979708097, −3.70275312779504616370534723278, −2.67453809492590823540517644386, −1.85762959976723200116325023109,
1.85762959976723200116325023109, 2.67453809492590823540517644386, 3.70275312779504616370534723278, 5.19489747629461411283979708097, 5.87828291361757467118536060459, 6.53349138395219407464163964289, 7.993353682382915666008091576358, 8.590644629105691064086774606872, 9.724681875952850212372395349151, 10.26781599779401054651658692830
