| L(s) = 1 | + 2.72·2-s − 2.69·3-s + 5.44·4-s + 0.220·5-s − 7.35·6-s + 2.22·7-s + 9.39·8-s + 4.27·9-s + 0.602·10-s − 5.25·11-s − 14.6·12-s + 5.85·13-s + 6.07·14-s − 0.595·15-s + 14.7·16-s − 17-s + 11.6·18-s − 1.11·19-s + 1.20·20-s − 6.00·21-s − 14.3·22-s + 6.93·23-s − 25.3·24-s − 4.95·25-s + 15.9·26-s − 3.43·27-s + 12.1·28-s + ⋯ |
| L(s) = 1 | + 1.92·2-s − 1.55·3-s + 2.72·4-s + 0.0987·5-s − 3.00·6-s + 0.841·7-s + 3.32·8-s + 1.42·9-s + 0.190·10-s − 1.58·11-s − 4.23·12-s + 1.62·13-s + 1.62·14-s − 0.153·15-s + 3.68·16-s − 0.242·17-s + 2.74·18-s − 0.255·19-s + 0.268·20-s − 1.31·21-s − 3.05·22-s + 1.44·23-s − 5.17·24-s − 0.990·25-s + 3.13·26-s − 0.660·27-s + 2.29·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.417531154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.417531154\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 - 2.72T + 2T^{2} \) |
| 3 | \( 1 + 2.69T + 3T^{2} \) |
| 5 | \( 1 - 0.220T + 5T^{2} \) |
| 7 | \( 1 - 2.22T + 7T^{2} \) |
| 11 | \( 1 + 5.25T + 11T^{2} \) |
| 13 | \( 1 - 5.85T + 13T^{2} \) |
| 19 | \( 1 + 1.11T + 19T^{2} \) |
| 23 | \( 1 - 6.93T + 23T^{2} \) |
| 29 | \( 1 - 8.50T + 29T^{2} \) |
| 31 | \( 1 + 6.37T + 31T^{2} \) |
| 37 | \( 1 - 3.29T + 37T^{2} \) |
| 41 | \( 1 + 4.85T + 41T^{2} \) |
| 47 | \( 1 + 6.19T + 47T^{2} \) |
| 53 | \( 1 - 3.51T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 + 5.26T + 61T^{2} \) |
| 67 | \( 1 - 3.14T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 + 1.28T + 73T^{2} \) |
| 79 | \( 1 - 0.258T + 79T^{2} \) |
| 83 | \( 1 + 9.75T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 - 1.61T + 97T^{2} \) |
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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
−10.93985333032529541496560263190, −10.34904983959057376662000404477, −8.286485068014767793960924982844, −7.26709115258529772047169981325, −6.33322758861829849109053767594, −5.69681703991264748844615380359, −5.02627342107751337136406857823, −4.39652611876229938066566580244, −3.04225204349673631005070108229, −1.54002373934081169102434898309,
1.54002373934081169102434898309, 3.04225204349673631005070108229, 4.39652611876229938066566580244, 5.02627342107751337136406857823, 5.69681703991264748844615380359, 6.33322758861829849109053767594, 7.26709115258529772047169981325, 8.286485068014767793960924982844, 10.34904983959057376662000404477, 10.93985333032529541496560263190
