| L(s) = 1 | + 4.70·2-s + 6.19·3-s + 14.1·4-s + 13.3·5-s + 29.1·6-s + 7·7-s + 29.0·8-s + 11.3·9-s + 63.0·10-s + 87.7·12-s + 59.1·13-s + 32.9·14-s + 82.9·15-s + 23.4·16-s − 47.9·17-s + 53.3·18-s + 39.4·19-s + 189.·20-s + 43.3·21-s − 100.·23-s + 179.·24-s + 54.3·25-s + 278.·26-s − 97.0·27-s + 99.1·28-s − 303.·29-s + 390.·30-s + ⋯ |
| L(s) = 1 | + 1.66·2-s + 1.19·3-s + 1.77·4-s + 1.19·5-s + 1.98·6-s + 0.377·7-s + 1.28·8-s + 0.419·9-s + 1.99·10-s + 2.11·12-s + 1.26·13-s + 0.629·14-s + 1.42·15-s + 0.365·16-s − 0.684·17-s + 0.698·18-s + 0.476·19-s + 2.12·20-s + 0.450·21-s − 0.911·23-s + 1.52·24-s + 0.435·25-s + 2.10·26-s − 0.691·27-s + 0.669·28-s − 1.94·29-s + 2.37·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(10.50045383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.50045383\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 4.70T + 8T^{2} \) |
| 3 | \( 1 - 6.19T + 27T^{2} \) |
| 5 | \( 1 - 13.3T + 125T^{2} \) |
| 13 | \( 1 - 59.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 47.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 39.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 100.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 303.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 45.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 215.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 522.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 347.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 513.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 163.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 653.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 595.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 293.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 361.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 245.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.14e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.07e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 612.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 612.T + 9.12e5T^{2} \) |
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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
−9.668052960006889253336440044254, −9.009985966797116223143246114822, −8.050167334362966500476036114596, −6.98222394877373161626528205279, −5.90661794754545675297722070852, −5.51874625423780523695346101815, −4.16101186765078900030992724682, −3.49331579732945962451835554429, −2.37454005265072005074171092437, −1.76704247533186477193243903046,
1.76704247533186477193243903046, 2.37454005265072005074171092437, 3.49331579732945962451835554429, 4.16101186765078900030992724682, 5.51874625423780523695346101815, 5.90661794754545675297722070852, 6.98222394877373161626528205279, 8.050167334362966500476036114596, 9.009985966797116223143246114822, 9.668052960006889253336440044254
