| L(s) = 1 | − 6.25·3-s + 17.1·5-s − 7·7-s + 12.1·9-s + 11·11-s + 51.7·13-s − 107.·15-s − 104.·17-s + 10.6·19-s + 43.7·21-s − 9.26·23-s + 168.·25-s + 93.0·27-s + 223.·29-s − 237.·31-s − 68.8·33-s − 119.·35-s + 361.·37-s − 323.·39-s + 273.·41-s + 149.·43-s + 207.·45-s + 531.·47-s + 49·49-s + 652.·51-s + 408.·53-s + 188.·55-s + ⋯ |
| L(s) = 1 | − 1.20·3-s + 1.53·5-s − 0.377·7-s + 0.448·9-s + 0.301·11-s + 1.10·13-s − 1.84·15-s − 1.48·17-s + 0.128·19-s + 0.454·21-s − 0.0840·23-s + 1.34·25-s + 0.663·27-s + 1.42·29-s − 1.37·31-s − 0.362·33-s − 0.578·35-s + 1.60·37-s − 1.32·39-s + 1.04·41-s + 0.530·43-s + 0.687·45-s + 1.65·47-s + 0.142·49-s + 1.79·51-s + 1.05·53-s + 0.461·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.533442083\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.533442083\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 - 11T \) |
| good | 3 | \( 1 + 6.25T + 27T^{2} \) |
| 5 | \( 1 - 17.1T + 125T^{2} \) |
| 13 | \( 1 - 51.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 104.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 10.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 9.26T + 1.21e4T^{2} \) |
| 29 | \( 1 - 223.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 237.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 361.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 273.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 149.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 531.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 408.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 416.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 393.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.04e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 552.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 724.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 516.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.39e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 329.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 10.6T + 9.12e5T^{2} \) |
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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
−11.04104259444640782290660147667, −10.57572954303012049253558579491, −9.459498804837724961463297965541, −8.724147461241812068621856503862, −6.90309451274365968659296679199, −6.11296939505751527923156979653, −5.64612566528288243171317581540, −4.31320658244832169830856285100, −2.44099995987620259662994753755, −0.940026695288567718687297924020,
0.940026695288567718687297924020, 2.44099995987620259662994753755, 4.31320658244832169830856285100, 5.64612566528288243171317581540, 6.11296939505751527923156979653, 6.90309451274365968659296679199, 8.724147461241812068621856503862, 9.459498804837724961463297965541, 10.57572954303012049253558579491, 11.04104259444640782290660147667
