| L(s) = 1 | − 81·3-s + 990·5-s + 8.57e3·7-s + 6.56e3·9-s + 7.05e4·11-s − 2.53e3·13-s − 8.01e4·15-s − 2.00e5·17-s − 6.95e5·19-s − 6.94e5·21-s + 2.47e6·23-s − 9.73e5·25-s − 5.31e5·27-s + 5.47e6·29-s + 3.73e6·31-s − 5.71e6·33-s + 8.49e6·35-s − 2.18e7·37-s + 2.04e5·39-s − 2.38e7·41-s + 1.06e7·43-s + 6.49e6·45-s + 2.39e6·47-s + 3.31e7·49-s + 1.62e7·51-s − 8.99e6·53-s + 6.98e7·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.708·5-s + 1.35·7-s + 1/3·9-s + 1.45·11-s − 0.0245·13-s − 0.408·15-s − 0.582·17-s − 1.22·19-s − 0.779·21-s + 1.84·23-s − 0.498·25-s − 0.192·27-s + 1.43·29-s + 0.725·31-s − 0.839·33-s + 0.956·35-s − 1.92·37-s + 0.0141·39-s − 1.31·41-s + 0.473·43-s + 0.236·45-s + 0.0716·47-s + 0.822·49-s + 0.336·51-s − 0.156·53-s + 1.02·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.718899344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.718899344\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 - 198 p T + p^{9} T^{2} \) |
| 7 | \( 1 - 8576 T + p^{9} T^{2} \) |
| 11 | \( 1 - 70596 T + p^{9} T^{2} \) |
| 13 | \( 1 + 2530 T + p^{9} T^{2} \) |
| 17 | \( 1 + 200574 T + p^{9} T^{2} \) |
| 19 | \( 1 + 695620 T + p^{9} T^{2} \) |
| 23 | \( 1 - 2472696 T + p^{9} T^{2} \) |
| 29 | \( 1 - 188766 p T + p^{9} T^{2} \) |
| 31 | \( 1 - 3732104 T + p^{9} T^{2} \) |
| 37 | \( 1 + 21898522 T + p^{9} T^{2} \) |
| 41 | \( 1 + 580950 p T + p^{9} T^{2} \) |
| 43 | \( 1 - 10612676 T + p^{9} T^{2} \) |
| 47 | \( 1 - 2398464 T + p^{9} T^{2} \) |
| 53 | \( 1 + 8994978 T + p^{9} T^{2} \) |
| 59 | \( 1 + 143417916 T + p^{9} T^{2} \) |
| 61 | \( 1 + 19804258 T + p^{9} T^{2} \) |
| 67 | \( 1 + 165625156 T + p^{9} T^{2} \) |
| 71 | \( 1 + 194801400 T + p^{9} T^{2} \) |
| 73 | \( 1 - 148729418 T + p^{9} T^{2} \) |
| 79 | \( 1 + 30134152 T + p^{9} T^{2} \) |
| 83 | \( 1 - 302054076 T + p^{9} T^{2} \) |
| 89 | \( 1 - 909502650 T + p^{9} T^{2} \) |
| 97 | \( 1 + 872463358 T + p^{9} T^{2} \) |
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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
−17.54119109031145543039992286201, −17.16725866894961141596132287952, −15.08530079699871765402691401359, −13.81065012044828245136095963270, −11.98277838239815058610830423027, −10.70457339204268290885214108209, −8.807800579608537854675563040325, −6.58000711696995112683251860089, −4.73731055289345091535714548756, −1.52345302993215290019211790749,
1.52345302993215290019211790749, 4.73731055289345091535714548756, 6.58000711696995112683251860089, 8.807800579608537854675563040325, 10.70457339204268290885214108209, 11.98277838239815058610830423027, 13.81065012044828245136095963270, 15.08530079699871765402691401359, 17.16725866894961141596132287952, 17.54119109031145543039992286201
