| L(s) = 1 | + 36·3-s − 3.49e3·5-s + 5.54e4·7-s − 1.75e5·9-s + 5.97e5·11-s + 1.37e6·13-s − 1.25e5·15-s + 1.01e7·17-s + 7.29e6·19-s + 1.99e6·21-s + 3.20e7·23-s − 3.66e7·25-s − 1.27e7·27-s − 1.36e7·29-s − 2.33e8·31-s + 2.14e7·33-s − 1.93e8·35-s − 2.57e8·37-s + 4.94e7·39-s − 2.21e8·41-s + 1.69e9·43-s + 6.13e8·45-s − 5.27e8·47-s + 1.09e9·49-s + 3.65e8·51-s + 3.27e9·53-s − 2.08e9·55-s + ⋯ |
| L(s) = 1 | + 0.0855·3-s − 0.499·5-s + 1.24·7-s − 0.992·9-s + 1.11·11-s + 1.02·13-s − 0.0427·15-s + 1.73·17-s + 0.676·19-s + 0.106·21-s + 1.03·23-s − 0.750·25-s − 0.170·27-s − 0.123·29-s − 1.46·31-s + 0.0955·33-s − 0.622·35-s − 0.611·37-s + 0.0877·39-s − 0.298·41-s + 1.76·43-s + 0.495·45-s − 0.335·47-s + 0.555·49-s + 0.148·51-s + 1.07·53-s − 0.558·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.995688736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.995688736\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 4 p^{2} T + p^{11} T^{2} \) |
| 5 | \( 1 + 698 p T + p^{11} T^{2} \) |
| 7 | \( 1 - 55464 T + p^{11} T^{2} \) |
| 11 | \( 1 - 597004 T + p^{11} T^{2} \) |
| 13 | \( 1 - 1373878 T + p^{11} T^{2} \) |
| 17 | \( 1 - 10140850 T + p^{11} T^{2} \) |
| 19 | \( 1 - 7297396 T + p^{11} T^{2} \) |
| 23 | \( 1 - 32057464 T + p^{11} T^{2} \) |
| 29 | \( 1 + 13605402 T + p^{11} T^{2} \) |
| 31 | \( 1 + 233160800 T + p^{11} T^{2} \) |
| 37 | \( 1 + 6967194 p T + p^{11} T^{2} \) |
| 41 | \( 1 + 221438598 T + p^{11} T^{2} \) |
| 43 | \( 1 - 1697758892 T + p^{11} T^{2} \) |
| 47 | \( 1 + 527509392 T + p^{11} T^{2} \) |
| 53 | \( 1 - 3277379822 T + p^{11} T^{2} \) |
| 59 | \( 1 - 3001908988 T + p^{11} T^{2} \) |
| 61 | \( 1 + 11630023610 T + p^{11} T^{2} \) |
| 67 | \( 1 - 17189000548 T + p^{11} T^{2} \) |
| 71 | \( 1 + 26169539608 T + p^{11} T^{2} \) |
| 73 | \( 1 + 7039021094 T + p^{11} T^{2} \) |
| 79 | \( 1 - 4199910416 T + p^{11} T^{2} \) |
| 83 | \( 1 - 39739936436 T + p^{11} T^{2} \) |
| 89 | \( 1 - 10565331594 T + p^{11} T^{2} \) |
| 97 | \( 1 + 69851645662 T + p^{11} T^{2} \) |
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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
−16.51799174268008002792390634138, −14.83462513723915397393947528455, −14.02626154197437452061755952217, −11.95306886219794136589820527124, −11.09052537705549742143964715254, −8.950974018986348614308245006273, −7.68638022615446622118461506358, −5.55423948923691221594204693898, −3.61021508107588038340171718716, −1.23915443595414167629153239699,
1.23915443595414167629153239699, 3.61021508107588038340171718716, 5.55423948923691221594204693898, 7.68638022615446622118461506358, 8.950974018986348614308245006273, 11.09052537705549742143964715254, 11.95306886219794136589820527124, 14.02626154197437452061755952217, 14.83462513723915397393947528455, 16.51799174268008002792390634138
