| L(s) = 1 | + 81·3-s + 830·5-s − 672·7-s + 6.56e3·9-s + 7.34e4·11-s − 7.82e4·13-s + 6.72e4·15-s − 1.61e5·17-s + 6.53e5·19-s − 5.44e4·21-s + 1.06e6·23-s − 1.26e6·25-s + 5.31e5·27-s + 3.82e6·29-s + 1.57e6·31-s + 5.95e6·33-s − 5.57e5·35-s + 1.60e7·37-s − 6.33e6·39-s + 2.62e7·41-s + 4.44e7·43-s + 5.44e6·45-s − 1.43e7·47-s − 3.99e7·49-s − 1.30e7·51-s − 2.43e7·53-s + 6.09e7·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.593·5-s − 0.105·7-s + 1/3·9-s + 1.51·11-s − 0.759·13-s + 0.342·15-s − 0.469·17-s + 1.15·19-s − 0.0610·21-s + 0.794·23-s − 0.647·25-s + 0.192·27-s + 1.00·29-s + 0.307·31-s + 0.873·33-s − 0.0628·35-s + 1.40·37-s − 0.438·39-s + 1.45·41-s + 1.98·43-s + 0.197·45-s − 0.428·47-s − 0.988·49-s − 0.271·51-s − 0.424·53-s + 0.898·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.872191781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.872191781\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{4} T \) |
| good | 5 | \( 1 - 166 p T + p^{9} T^{2} \) |
| 7 | \( 1 + 96 p T + p^{9} T^{2} \) |
| 11 | \( 1 - 73468 T + p^{9} T^{2} \) |
| 13 | \( 1 + 78242 T + p^{9} T^{2} \) |
| 17 | \( 1 + 161726 T + p^{9} T^{2} \) |
| 19 | \( 1 - 653572 T + p^{9} T^{2} \) |
| 23 | \( 1 - 1066696 T + p^{9} T^{2} \) |
| 29 | \( 1 - 3824838 T + p^{9} T^{2} \) |
| 31 | \( 1 - 1579480 T + p^{9} T^{2} \) |
| 37 | \( 1 - 16015590 T + p^{9} T^{2} \) |
| 41 | \( 1 - 26268282 T + p^{9} T^{2} \) |
| 43 | \( 1 - 44495228 T + p^{9} T^{2} \) |
| 47 | \( 1 + 14324160 T + p^{9} T^{2} \) |
| 53 | \( 1 + 24386050 T + p^{9} T^{2} \) |
| 59 | \( 1 + 11942084 T + p^{9} T^{2} \) |
| 61 | \( 1 + 189740258 T + p^{9} T^{2} \) |
| 67 | \( 1 - 106709572 T + p^{9} T^{2} \) |
| 71 | \( 1 + 302754376 T + p^{9} T^{2} \) |
| 73 | \( 1 - 81769546 T + p^{9} T^{2} \) |
| 79 | \( 1 + 315315352 T + p^{9} T^{2} \) |
| 83 | \( 1 + 752833276 T + p^{9} T^{2} \) |
| 89 | \( 1 + 433284294 T + p^{9} T^{2} \) |
| 97 | \( 1 - 1282496642 T + p^{9} T^{2} \) |
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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
−13.91502407016996434685990361901, −12.64205307230898884402997463629, −11.40911921235644863493336524144, −9.764456302600960412951207998332, −9.098101625762862443300483946574, −7.47804394476346696627245687695, −6.16459118159754917706414402033, −4.40046957515917179677842838693, −2.76775299823414959572029204998, −1.21118680462325605242876891964,
1.21118680462325605242876891964, 2.76775299823414959572029204998, 4.40046957515917179677842838693, 6.16459118159754917706414402033, 7.47804394476346696627245687695, 9.098101625762862443300483946574, 9.764456302600960412951207998332, 11.40911921235644863493336524144, 12.64205307230898884402997463629, 13.91502407016996434685990361901
