L(s) = 1 | − 2-s − 1.96·3-s + 4-s + 1.55·5-s + 1.96·6-s − 2.12·7-s − 8-s + 0.845·9-s − 1.55·10-s + 3.46·11-s − 1.96·12-s + 3.94·13-s + 2.12·14-s − 3.04·15-s + 16-s + 6.42·17-s − 0.845·18-s + 2.45·19-s + 1.55·20-s + 4.16·21-s − 3.46·22-s + 8.71·23-s + 1.96·24-s − 2.58·25-s − 3.94·26-s + 4.22·27-s − 2.12·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.13·3-s + 0.5·4-s + 0.694·5-s + 0.800·6-s − 0.802·7-s − 0.353·8-s + 0.281·9-s − 0.491·10-s + 1.04·11-s − 0.566·12-s + 1.09·13-s + 0.567·14-s − 0.786·15-s + 0.250·16-s + 1.55·17-s − 0.199·18-s + 0.562·19-s + 0.347·20-s + 0.908·21-s − 0.738·22-s + 1.81·23-s + 0.400·24-s − 0.517·25-s − 0.774·26-s + 0.813·27-s − 0.401·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.153525483\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.153525483\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
good | 3 | \( 1 + 1.96T + 3T^{2} \) |
| 5 | \( 1 - 1.55T + 5T^{2} \) |
| 7 | \( 1 + 2.12T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 - 3.94T + 13T^{2} \) |
| 17 | \( 1 - 6.42T + 17T^{2} \) |
| 19 | \( 1 - 2.45T + 19T^{2} \) |
| 23 | \( 1 - 8.71T + 23T^{2} \) |
| 29 | \( 1 - 0.0411T + 29T^{2} \) |
| 31 | \( 1 - 2.72T + 31T^{2} \) |
| 37 | \( 1 + 0.677T + 37T^{2} \) |
| 41 | \( 1 + 1.96T + 41T^{2} \) |
| 43 | \( 1 - 1.87T + 43T^{2} \) |
| 47 | \( 1 + 3.79T + 47T^{2} \) |
| 53 | \( 1 + 4.51T + 53T^{2} \) |
| 59 | \( 1 - 3.27T + 59T^{2} \) |
| 61 | \( 1 + 5.93T + 61T^{2} \) |
| 67 | \( 1 + 4.52T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 + 3.46T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 7.79T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + 2.89T + 97T^{2} \) |
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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
−8.576470162741657904187760311237, −7.66867548715033776792783377184, −6.64672768378586159656656251887, −6.38939554724051156951046803682, −5.67642610534243431817036102306, −5.03352545429230077137717591039, −3.64122540004739487285247636067, −2.98730621250791011409078974612, −1.45296192361437178750496072557, −0.814458876710630688456281634482,
0.814458876710630688456281634482, 1.45296192361437178750496072557, 2.98730621250791011409078974612, 3.64122540004739487285247636067, 5.03352545429230077137717591039, 5.67642610534243431817036102306, 6.38939554724051156951046803682, 6.64672768378586159656656251887, 7.66867548715033776792783377184, 8.576470162741657904187760311237