| L(s) = 1 | + 9.92·2-s − 0.133·3-s + 66.4·4-s − 25·5-s − 1.32·6-s + 342.·8-s − 242.·9-s − 248.·10-s + 674.·11-s − 8.87·12-s + 770.·13-s + 3.33·15-s + 1.26e3·16-s + 693.·17-s − 2.41e3·18-s + 1.39e3·19-s − 1.66e3·20-s + 6.69e3·22-s + 1.32e3·23-s − 45.6·24-s + 625·25-s + 7.64e3·26-s + 64.8·27-s + 172.·29-s + 33.1·30-s + 4.69e3·31-s + 1.64e3·32-s + ⋯ |
| L(s) = 1 | + 1.75·2-s − 0.00855·3-s + 2.07·4-s − 0.447·5-s − 0.0150·6-s + 1.89·8-s − 0.999·9-s − 0.784·10-s + 1.68·11-s − 0.0177·12-s + 1.26·13-s + 0.00382·15-s + 1.23·16-s + 0.582·17-s − 1.75·18-s + 0.884·19-s − 0.929·20-s + 2.94·22-s + 0.521·23-s − 0.0161·24-s + 0.200·25-s + 2.21·26-s + 0.0171·27-s + 0.0381·29-s + 0.00671·30-s + 0.877·31-s + 0.283·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.163873167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.163873167\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 9.92T + 32T^{2} \) |
| 3 | \( 1 + 0.133T + 243T^{2} \) |
| 11 | \( 1 - 674.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 770.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 693.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.39e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.32e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 172.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.69e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.32e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.07e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.54e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.37e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 6.72e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.85e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 7.67e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.90e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.27e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.28e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.01e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.08e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.66e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.47e5T + 8.58e9T^{2} \) |
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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
−11.82966887035331928116005564734, −10.87613389169133876700276631567, −9.252008334677060245429142667653, −8.087537396209143030218421210748, −6.67407323845330201143446017470, −6.05394756225908670305503749340, −4.90949403634112374454775352294, −3.73503843198901761723480456622, −3.09863103128117729442890770989, −1.28792565574404015322352352708,
1.28792565574404015322352352708, 3.09863103128117729442890770989, 3.73503843198901761723480456622, 4.90949403634112374454775352294, 6.05394756225908670305503749340, 6.67407323845330201143446017470, 8.087537396209143030218421210748, 9.252008334677060245429142667653, 10.87613389169133876700276631567, 11.82966887035331928116005564734
