| L(s) = 1 | + 243·3-s − 5.37e3·5-s + 2.77e4·7-s + 5.90e4·9-s − 6.37e5·11-s + 7.66e5·13-s − 1.30e6·15-s + 3.08e6·17-s + 1.95e7·19-s + 6.74e6·21-s − 1.53e7·23-s − 1.99e7·25-s + 1.43e7·27-s + 1.07e7·29-s + 5.09e7·31-s − 1.54e8·33-s − 1.49e8·35-s + 6.64e8·37-s + 1.86e8·39-s + 8.98e8·41-s + 9.57e8·43-s − 3.17e8·45-s + 1.55e9·47-s − 1.20e9·49-s + 7.49e8·51-s + 3.79e9·53-s + 3.42e9·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.768·5-s + 0.624·7-s + 1/3·9-s − 1.19·11-s + 0.572·13-s − 0.443·15-s + 0.526·17-s + 1.80·19-s + 0.360·21-s − 0.496·23-s − 0.409·25-s + 0.192·27-s + 0.0973·29-s + 0.319·31-s − 0.689·33-s − 0.479·35-s + 1.57·37-s + 0.330·39-s + 1.21·41-s + 0.993·43-s − 0.256·45-s + 0.989·47-s − 0.610·49-s + 0.304·51-s + 1.24·53-s + 0.917·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.329704194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.329704194\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{5} T \) |
| good | 5 | \( 1 + 1074 p T + p^{11} T^{2} \) |
| 7 | \( 1 - 27760 T + p^{11} T^{2} \) |
| 11 | \( 1 + 637836 T + p^{11} T^{2} \) |
| 13 | \( 1 - 766214 T + p^{11} T^{2} \) |
| 17 | \( 1 - 3084354 T + p^{11} T^{2} \) |
| 19 | \( 1 - 1026916 p T + p^{11} T^{2} \) |
| 23 | \( 1 + 15312360 T + p^{11} T^{2} \) |
| 29 | \( 1 - 10751262 T + p^{11} T^{2} \) |
| 31 | \( 1 - 50937400 T + p^{11} T^{2} \) |
| 37 | \( 1 - 664740830 T + p^{11} T^{2} \) |
| 41 | \( 1 - 898833450 T + p^{11} T^{2} \) |
| 43 | \( 1 - 957947188 T + p^{11} T^{2} \) |
| 47 | \( 1 - 1555741344 T + p^{11} T^{2} \) |
| 53 | \( 1 - 3792417030 T + p^{11} T^{2} \) |
| 59 | \( 1 + 555306924 T + p^{11} T^{2} \) |
| 61 | \( 1 - 4950420998 T + p^{11} T^{2} \) |
| 67 | \( 1 + 5292399284 T + p^{11} T^{2} \) |
| 71 | \( 1 - 14831086248 T + p^{11} T^{2} \) |
| 73 | \( 1 - 13971005210 T + p^{11} T^{2} \) |
| 79 | \( 1 + 3720542360 T + p^{11} T^{2} \) |
| 83 | \( 1 + 8768454036 T + p^{11} T^{2} \) |
| 89 | \( 1 + 25472769174 T + p^{11} T^{2} \) |
| 97 | \( 1 + 39092494846 T + p^{11} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.38747653270288016852068230178, −12.05695940691086763281043807732, −10.97588864844907187308039000207, −9.622493494060053533760370251564, −8.092417492814132081515676744450, −7.56752554715242147898814321489, −5.51104660420037696357227831350, −4.03589566754523705259203314360, −2.67841354440558860630189672304, −0.926088275589403076715102498089,
0.926088275589403076715102498089, 2.67841354440558860630189672304, 4.03589566754523705259203314360, 5.51104660420037696357227831350, 7.56752554715242147898814321489, 8.092417492814132081515676744450, 9.622493494060053533760370251564, 10.97588864844907187308039000207, 12.05695940691086763281043807732, 13.38747653270288016852068230178
