| L(s) = 1 | + 2.14·2-s + 2.53·3-s + 2.59·4-s + 5-s + 5.44·6-s − 1.53·7-s + 1.27·8-s + 3.44·9-s + 2.14·10-s + 11-s + 6.58·12-s − 1.14·13-s − 3.29·14-s + 2.53·15-s − 2.45·16-s + 3.14·17-s + 7.39·18-s + 19-s + 2.59·20-s − 3.90·21-s + 2.14·22-s − 5.10·23-s + 3.23·24-s + 25-s − 2.45·26-s + 1.13·27-s − 3.98·28-s + ⋯ |
| L(s) = 1 | + 1.51·2-s + 1.46·3-s + 1.29·4-s + 0.447·5-s + 2.22·6-s − 0.580·7-s + 0.450·8-s + 1.14·9-s + 0.677·10-s + 0.301·11-s + 1.90·12-s − 0.317·13-s − 0.879·14-s + 0.655·15-s − 0.614·16-s + 0.762·17-s + 1.74·18-s + 0.229·19-s + 0.580·20-s − 0.851·21-s + 0.456·22-s − 1.06·23-s + 0.660·24-s + 0.200·25-s − 0.481·26-s + 0.219·27-s − 0.752·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.635062614\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.635062614\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.14T + 2T^{2} \) |
| 3 | \( 1 - 2.53T + 3T^{2} \) |
| 7 | \( 1 + 1.53T + 7T^{2} \) |
| 13 | \( 1 + 1.14T + 13T^{2} \) |
| 17 | \( 1 - 3.14T + 17T^{2} \) |
| 23 | \( 1 + 5.10T + 23T^{2} \) |
| 29 | \( 1 - 2.99T + 29T^{2} \) |
| 31 | \( 1 - 0.460T + 31T^{2} \) |
| 37 | \( 1 + 7.62T + 37T^{2} \) |
| 41 | \( 1 + 1.79T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 + 0.0901T + 47T^{2} \) |
| 53 | \( 1 - 2.98T + 53T^{2} \) |
| 59 | \( 1 + 7.98T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 - 9.20T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 4.27T + 79T^{2} \) |
| 83 | \( 1 + 3.17T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 + 17.1T + 97T^{2} \) |
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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
−9.732415549832508041921449201633, −9.199989053254608198946554382690, −8.207648034109615999676147819179, −7.28273500768509868964506779519, −6.36513368387485277847101141684, −5.53944619223417438689561607864, −4.43398778161524492062503848270, −3.53203242998790049031314750370, −2.92592312187510848208571032282, −1.94821075852507817297597050519,
1.94821075852507817297597050519, 2.92592312187510848208571032282, 3.53203242998790049031314750370, 4.43398778161524492062503848270, 5.53944619223417438689561607864, 6.36513368387485277847101141684, 7.28273500768509868964506779519, 8.207648034109615999676147819179, 9.199989053254608198946554382690, 9.732415549832508041921449201633
