| L(s) = 1 | + 2.94·3-s − 5-s − 2.13·7-s + 5.69·9-s + 5.67·11-s + 5.04·13-s − 2.94·15-s + 0.545·17-s + 6.74·19-s − 6.30·21-s + 23-s + 25-s + 7.93·27-s + 5.11·29-s − 4.64·31-s + 16.7·33-s + 2.13·35-s + 2.21·37-s + 14.8·39-s − 10.2·41-s − 10.5·43-s − 5.69·45-s + 2.61·47-s − 2.42·49-s + 1.60·51-s − 8.67·53-s − 5.67·55-s + ⋯ |
| L(s) = 1 | + 1.70·3-s − 0.447·5-s − 0.808·7-s + 1.89·9-s + 1.71·11-s + 1.39·13-s − 0.761·15-s + 0.132·17-s + 1.54·19-s − 1.37·21-s + 0.208·23-s + 0.200·25-s + 1.52·27-s + 0.949·29-s − 0.834·31-s + 2.91·33-s + 0.361·35-s + 0.364·37-s + 2.38·39-s − 1.59·41-s − 1.60·43-s − 0.848·45-s + 0.381·47-s − 0.346·49-s + 0.225·51-s − 1.19·53-s − 0.764·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.362343894\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.362343894\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 2.94T + 3T^{2} \) |
| 7 | \( 1 + 2.13T + 7T^{2} \) |
| 11 | \( 1 - 5.67T + 11T^{2} \) |
| 13 | \( 1 - 5.04T + 13T^{2} \) |
| 17 | \( 1 - 0.545T + 17T^{2} \) |
| 19 | \( 1 - 6.74T + 19T^{2} \) |
| 29 | \( 1 - 5.11T + 29T^{2} \) |
| 31 | \( 1 + 4.64T + 31T^{2} \) |
| 37 | \( 1 - 2.21T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 - 2.61T + 47T^{2} \) |
| 53 | \( 1 + 8.67T + 53T^{2} \) |
| 59 | \( 1 + 0.777T + 59T^{2} \) |
| 61 | \( 1 - 7.00T + 61T^{2} \) |
| 67 | \( 1 - 7.27T + 67T^{2} \) |
| 71 | \( 1 - 3.40T + 71T^{2} \) |
| 73 | \( 1 + 6.34T + 73T^{2} \) |
| 79 | \( 1 + 4.07T + 79T^{2} \) |
| 83 | \( 1 - 1.51T + 83T^{2} \) |
| 89 | \( 1 + 9.79T + 89T^{2} \) |
| 97 | \( 1 - 4.60T + 97T^{2} \) |
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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
−8.135838978755787295887611114540, −7.19454127337660350641189190267, −6.76325086496967537859952641950, −6.04085021418037159712588986400, −4.83860573517296086050544737239, −3.88196667687638137324562592481, −3.39992662346822163008425334893, −3.14248803111852525258980027920, −1.76965705102324725335080182389, −1.07316626794452437790630706547,
1.07316626794452437790630706547, 1.76965705102324725335080182389, 3.14248803111852525258980027920, 3.39992662346822163008425334893, 3.88196667687638137324562592481, 4.83860573517296086050544737239, 6.04085021418037159712588986400, 6.76325086496967537859952641950, 7.19454127337660350641189190267, 8.135838978755787295887611114540
