| L(s) = 1 | + 1.77·3-s + 80.6·5-s + 236.·7-s − 239.·9-s − 634.·11-s + 374.·13-s + 142.·15-s + 990.·17-s + 2.96e3·19-s + 419.·21-s − 529·23-s + 3.38e3·25-s − 854.·27-s − 784.·29-s + 3.77e3·31-s − 1.12e3·33-s + 1.90e4·35-s − 4.50e3·37-s + 662.·39-s − 7.58e3·41-s + 6.51e3·43-s − 1.93e4·45-s − 8.90e3·47-s + 3.92e4·49-s + 1.75e3·51-s + 1.17e3·53-s − 5.12e4·55-s + ⋯ |
| L(s) = 1 | + 0.113·3-s + 1.44·5-s + 1.82·7-s − 0.987·9-s − 1.58·11-s + 0.614·13-s + 0.163·15-s + 0.831·17-s + 1.88·19-s + 0.207·21-s − 0.208·23-s + 1.08·25-s − 0.225·27-s − 0.173·29-s + 0.706·31-s − 0.179·33-s + 2.63·35-s − 0.541·37-s + 0.0697·39-s − 0.705·41-s + 0.537·43-s − 1.42·45-s − 0.587·47-s + 2.33·49-s + 0.0943·51-s + 0.0576·53-s − 2.28·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.665091240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.665091240\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + 529T \) |
| good | 3 | \( 1 - 1.77T + 243T^{2} \) |
| 5 | \( 1 - 80.6T + 3.12e3T^{2} \) |
| 7 | \( 1 - 236.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 634.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 374.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 990.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.96e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 784.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.77e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.50e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.58e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.51e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 8.90e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.17e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.88e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.21e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.42e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.13e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.74e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.60e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.79e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.06e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.12e4T + 8.58e9T^{2} \) |
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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
−13.55721467313517134751474027832, −11.89472478067585804108828950234, −10.89543169958750374650069618905, −9.912314601653357142722744074735, −8.539285334484289554159995492114, −7.67978816795963632942594731031, −5.61826898647124550696559629666, −5.21339087819810365207902315072, −2.75515942074681724524511972563, −1.41912085942650671140141303815,
1.41912085942650671140141303815, 2.75515942074681724524511972563, 5.21339087819810365207902315072, 5.61826898647124550696559629666, 7.67978816795963632942594731031, 8.539285334484289554159995492114, 9.912314601653357142722744074735, 10.89543169958750374650069618905, 11.89472478067585804108828950234, 13.55721467313517134751474027832
