| L(s) = 1 | + 1.68·2-s − 3.22·3-s + 0.849·4-s + 5-s − 5.44·6-s − 2.35·7-s − 1.94·8-s + 7.40·9-s + 1.68·10-s − 11-s − 2.74·12-s − 6.06·13-s − 3.96·14-s − 3.22·15-s − 4.97·16-s + 6.39·17-s + 12.5·18-s − 6.75·19-s + 0.849·20-s + 7.58·21-s − 1.68·22-s − 6.45·23-s + 6.26·24-s + 25-s − 10.2·26-s − 14.2·27-s − 1.99·28-s + ⋯ |
| L(s) = 1 | + 1.19·2-s − 1.86·3-s + 0.424·4-s + 0.447·5-s − 2.22·6-s − 0.888·7-s − 0.686·8-s + 2.46·9-s + 0.533·10-s − 0.301·11-s − 0.791·12-s − 1.68·13-s − 1.06·14-s − 0.833·15-s − 1.24·16-s + 1.55·17-s + 2.94·18-s − 1.54·19-s + 0.190·20-s + 1.65·21-s − 0.359·22-s − 1.34·23-s + 1.27·24-s + 0.200·25-s − 2.00·26-s − 2.73·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7981816513\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7981816513\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 - 1.68T + 2T^{2} \) |
| 3 | \( 1 + 3.22T + 3T^{2} \) |
| 7 | \( 1 + 2.35T + 7T^{2} \) |
| 13 | \( 1 + 6.06T + 13T^{2} \) |
| 17 | \( 1 - 6.39T + 17T^{2} \) |
| 19 | \( 1 + 6.75T + 19T^{2} \) |
| 23 | \( 1 + 6.45T + 23T^{2} \) |
| 29 | \( 1 - 0.579T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 - 0.543T + 37T^{2} \) |
| 41 | \( 1 + 4.60T + 41T^{2} \) |
| 43 | \( 1 + 3.33T + 43T^{2} \) |
| 47 | \( 1 + 1.45T + 47T^{2} \) |
| 53 | \( 1 - 3.33T + 53T^{2} \) |
| 59 | \( 1 - 7.22T + 59T^{2} \) |
| 61 | \( 1 + 4.24T + 61T^{2} \) |
| 67 | \( 1 - 3.36T + 67T^{2} \) |
| 71 | \( 1 + 1.15T + 71T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 + 7.30T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 3.84T + 97T^{2} \) |
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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
−8.296443580340958587952096218809, −7.22018305808921629923460826519, −6.52300921875457571145164972910, −6.04668231704048735293613137666, −5.49504315825800466072603413491, −4.77910509844430491566443562565, −4.28758418376475052006554332351, −3.17670774502970962244762110956, −2.10406190080160627760092379558, −0.44482818649967276695625400280,
0.44482818649967276695625400280, 2.10406190080160627760092379558, 3.17670774502970962244762110956, 4.28758418376475052006554332351, 4.77910509844430491566443562565, 5.49504315825800466072603413491, 6.04668231704048735293613137666, 6.52300921875457571145164972910, 7.22018305808921629923460826519, 8.296443580340958587952096218809
