| L(s) = 1 | − 1.92·2-s + 0.431·3-s + 1.70·4-s − 5-s − 0.831·6-s − 3.01·7-s + 0.564·8-s − 2.81·9-s + 1.92·10-s + 0.386·11-s + 0.736·12-s + 6.99·13-s + 5.81·14-s − 0.431·15-s − 4.50·16-s + 6.58·17-s + 5.41·18-s − 5.57·19-s − 1.70·20-s − 1.30·21-s − 0.744·22-s + 2.83·23-s + 0.243·24-s + 25-s − 13.4·26-s − 2.50·27-s − 5.15·28-s + ⋯ |
| L(s) = 1 | − 1.36·2-s + 0.249·3-s + 0.853·4-s − 0.447·5-s − 0.339·6-s − 1.14·7-s + 0.199·8-s − 0.937·9-s + 0.608·10-s + 0.116·11-s + 0.212·12-s + 1.93·13-s + 1.55·14-s − 0.111·15-s − 1.12·16-s + 1.59·17-s + 1.27·18-s − 1.27·19-s − 0.381·20-s − 0.284·21-s − 0.158·22-s + 0.590·23-s + 0.0497·24-s + 0.200·25-s − 2.64·26-s − 0.483·27-s − 0.974·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5901778973\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5901778973\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 83 | \( 1 - T \) |
| good | 2 | \( 1 + 1.92T + 2T^{2} \) |
| 3 | \( 1 - 0.431T + 3T^{2} \) |
| 7 | \( 1 + 3.01T + 7T^{2} \) |
| 11 | \( 1 - 0.386T + 11T^{2} \) |
| 13 | \( 1 - 6.99T + 13T^{2} \) |
| 17 | \( 1 - 6.58T + 17T^{2} \) |
| 19 | \( 1 + 5.57T + 19T^{2} \) |
| 23 | \( 1 - 2.83T + 23T^{2} \) |
| 29 | \( 1 - 6.46T + 29T^{2} \) |
| 31 | \( 1 - 3.65T + 31T^{2} \) |
| 37 | \( 1 + 2.48T + 37T^{2} \) |
| 41 | \( 1 - 8.52T + 41T^{2} \) |
| 43 | \( 1 - 6.95T + 43T^{2} \) |
| 47 | \( 1 - 1.08T + 47T^{2} \) |
| 53 | \( 1 - 5.27T + 53T^{2} \) |
| 59 | \( 1 - 4.70T + 59T^{2} \) |
| 61 | \( 1 - 5.05T + 61T^{2} \) |
| 67 | \( 1 - 4.19T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 - 7.75T + 79T^{2} \) |
| 89 | \( 1 - 0.0634T + 89T^{2} \) |
| 97 | \( 1 - 6.29T + 97T^{2} \) |
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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
−10.86551071520328339278503636595, −10.25729679590072721353303735766, −9.173026646809153716437892793753, −8.588948237210596065313392641981, −7.929952812210624405140238297031, −6.69886724721982666414785424496, −5.86502736023281965132951391907, −3.98177901227239708484002990655, −2.88301438185525796574743955242, −0.898941806920630580341991713219,
0.898941806920630580341991713219, 2.88301438185525796574743955242, 3.98177901227239708484002990655, 5.86502736023281965132951391907, 6.69886724721982666414785424496, 7.929952812210624405140238297031, 8.588948237210596065313392641981, 9.173026646809153716437892793753, 10.25729679590072721353303735766, 10.86551071520328339278503636595
