| L(s) = 1 | + 3-s − 5-s − 2.96·7-s + 9-s + 4.54·11-s − 7.02·13-s − 15-s + 5.76·17-s − 2.05·19-s − 2.96·21-s + 0.736·23-s + 25-s + 27-s + 3.25·29-s + 31-s + 4.54·33-s + 2.96·35-s + 1.09·37-s − 7.02·39-s − 0.440·41-s − 4.49·43-s − 45-s − 5.27·47-s + 1.77·49-s + 5.76·51-s + 11.7·53-s − 4.54·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.11·7-s + 0.333·9-s + 1.36·11-s − 1.94·13-s − 0.258·15-s + 1.39·17-s − 0.471·19-s − 0.646·21-s + 0.153·23-s + 0.200·25-s + 0.192·27-s + 0.605·29-s + 0.179·31-s + 0.790·33-s + 0.500·35-s + 0.179·37-s − 1.12·39-s − 0.0688·41-s − 0.685·43-s − 0.149·45-s − 0.769·47-s + 0.254·49-s + 0.806·51-s + 1.61·53-s − 0.612·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.778246260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.778246260\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 2.96T + 7T^{2} \) |
| 11 | \( 1 - 4.54T + 11T^{2} \) |
| 13 | \( 1 + 7.02T + 13T^{2} \) |
| 17 | \( 1 - 5.76T + 17T^{2} \) |
| 19 | \( 1 + 2.05T + 19T^{2} \) |
| 23 | \( 1 - 0.736T + 23T^{2} \) |
| 29 | \( 1 - 3.25T + 29T^{2} \) |
| 37 | \( 1 - 1.09T + 37T^{2} \) |
| 41 | \( 1 + 0.440T + 41T^{2} \) |
| 43 | \( 1 + 4.49T + 43T^{2} \) |
| 47 | \( 1 + 5.27T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 - 3.74T + 59T^{2} \) |
| 61 | \( 1 - 3.51T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 + 4.07T + 73T^{2} \) |
| 79 | \( 1 - 4.06T + 79T^{2} \) |
| 83 | \( 1 + 3.80T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 + 1.48T + 97T^{2} \) |
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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
−8.503553165863995628321507660019, −7.79312531657775233735719271818, −6.93556958532504804423842117587, −6.63862635593225973308858856069, −5.48892812870433852270130448027, −4.58405995509532895966011109131, −3.71577321708172761137896204276, −3.11141233250029176863144926494, −2.16388069779699228386709754003, −0.74400980576831977973382701753,
0.74400980576831977973382701753, 2.16388069779699228386709754003, 3.11141233250029176863144926494, 3.71577321708172761137896204276, 4.58405995509532895966011109131, 5.48892812870433852270130448027, 6.63862635593225973308858856069, 6.93556958532504804423842117587, 7.79312531657775233735719271818, 8.503553165863995628321507660019
