| L(s) = 1 | + 0.115·2-s − 1.98·4-s − 5-s + 7-s − 0.460·8-s − 0.115·10-s − 1.58·11-s + 4.07·13-s + 0.115·14-s + 3.92·16-s − 1.16·17-s − 7.51·19-s + 1.98·20-s − 0.182·22-s + 23-s + 25-s + 0.470·26-s − 1.98·28-s − 4.41·29-s + 0.445·31-s + 1.37·32-s − 0.134·34-s − 35-s + 11.6·37-s − 0.867·38-s + 0.460·40-s − 0.639·41-s + ⋯ |
| L(s) = 1 | + 0.0816·2-s − 0.993·4-s − 0.447·5-s + 0.377·7-s − 0.162·8-s − 0.0365·10-s − 0.476·11-s + 1.13·13-s + 0.0308·14-s + 0.980·16-s − 0.281·17-s − 1.72·19-s + 0.444·20-s − 0.0389·22-s + 0.208·23-s + 0.200·25-s + 0.0923·26-s − 0.375·28-s − 0.820·29-s + 0.0800·31-s + 0.242·32-s − 0.0229·34-s − 0.169·35-s + 1.90·37-s − 0.140·38-s + 0.0727·40-s − 0.0999·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.125786977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.125786977\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 0.115T + 2T^{2} \) |
| 11 | \( 1 + 1.58T + 11T^{2} \) |
| 13 | \( 1 - 4.07T + 13T^{2} \) |
| 17 | \( 1 + 1.16T + 17T^{2} \) |
| 19 | \( 1 + 7.51T + 19T^{2} \) |
| 29 | \( 1 + 4.41T + 29T^{2} \) |
| 31 | \( 1 - 0.445T + 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 + 0.639T + 41T^{2} \) |
| 43 | \( 1 + 5.64T + 43T^{2} \) |
| 47 | \( 1 + 1.16T + 47T^{2} \) |
| 53 | \( 1 + 3.22T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 + 3.64T + 61T^{2} \) |
| 67 | \( 1 - 6.81T + 67T^{2} \) |
| 71 | \( 1 + 14.8T + 71T^{2} \) |
| 73 | \( 1 + 2.40T + 73T^{2} \) |
| 79 | \( 1 + 9.07T + 79T^{2} \) |
| 83 | \( 1 + 3.52T + 83T^{2} \) |
| 89 | \( 1 + 6.27T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−8.103470542608668984399502211447, −7.37774229999136474158937101930, −6.37605627102440877278913500493, −5.81845198172594921224246753495, −4.94543518410280520914237047199, −4.28362454187047071336819258822, −3.81112034458382227753992395873, −2.83270294348075955391582088298, −1.70195437636835559569697146870, −0.54013221939296004569194108611,
0.54013221939296004569194108611, 1.70195437636835559569697146870, 2.83270294348075955391582088298, 3.81112034458382227753992395873, 4.28362454187047071336819258822, 4.94543518410280520914237047199, 5.81845198172594921224246753495, 6.37605627102440877278913500493, 7.37774229999136474158937101930, 8.103470542608668984399502211447
