| L(s) = 1 | + 2.44·2-s − 1.26·3-s + 3.98·4-s + 4.00·5-s − 3.09·6-s − 7-s + 4.84·8-s − 1.39·9-s + 9.79·10-s − 5.33·11-s − 5.03·12-s + 4.30·13-s − 2.44·14-s − 5.06·15-s + 3.88·16-s − 3.56·17-s − 3.42·18-s + 0.0482·19-s + 15.9·20-s + 1.26·21-s − 13.0·22-s + 2.49·23-s − 6.12·24-s + 11.0·25-s + 10.5·26-s + 5.56·27-s − 3.98·28-s + ⋯ |
| L(s) = 1 | + 1.72·2-s − 0.730·3-s + 1.99·4-s + 1.79·5-s − 1.26·6-s − 0.377·7-s + 1.71·8-s − 0.466·9-s + 3.09·10-s − 1.60·11-s − 1.45·12-s + 1.19·13-s − 0.653·14-s − 1.30·15-s + 0.971·16-s − 0.864·17-s − 0.806·18-s + 0.0110·19-s + 3.56·20-s + 0.276·21-s − 2.78·22-s + 0.520·23-s − 1.25·24-s + 2.21·25-s + 2.06·26-s + 1.07·27-s − 0.752·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.185401463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.185401463\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| good | 2 | \( 1 - 2.44T + 2T^{2} \) |
| 3 | \( 1 + 1.26T + 3T^{2} \) |
| 5 | \( 1 - 4.00T + 5T^{2} \) |
| 11 | \( 1 + 5.33T + 11T^{2} \) |
| 13 | \( 1 - 4.30T + 13T^{2} \) |
| 17 | \( 1 + 3.56T + 17T^{2} \) |
| 19 | \( 1 - 0.0482T + 19T^{2} \) |
| 23 | \( 1 - 2.49T + 23T^{2} \) |
| 29 | \( 1 + 7.78T + 29T^{2} \) |
| 31 | \( 1 - 0.0115T + 31T^{2} \) |
| 37 | \( 1 - 8.32T + 37T^{2} \) |
| 41 | \( 1 + 3.81T + 41T^{2} \) |
| 43 | \( 1 + 12.4T + 43T^{2} \) |
| 47 | \( 1 + 0.534T + 47T^{2} \) |
| 59 | \( 1 - 7.90T + 59T^{2} \) |
| 61 | \( 1 - 4.56T + 61T^{2} \) |
| 67 | \( 1 + 8.10T + 67T^{2} \) |
| 71 | \( 1 - 6.98T + 71T^{2} \) |
| 73 | \( 1 - 0.544T + 73T^{2} \) |
| 79 | \( 1 + 3.77T + 79T^{2} \) |
| 83 | \( 1 + 3.56T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.37590417769541719802945222772, −10.87318023524327909627572693642, −9.953219126276000935064581988431, −8.647701032249732197105625060814, −6.88889644550852025899291596638, −6.04568926384736410476689029063, −5.60001704202704564566319630582, −4.85364028630797150545842473008, −3.15648140586513172130957996836, −2.14363193338510574787657531576,
2.14363193338510574787657531576, 3.15648140586513172130957996836, 4.85364028630797150545842473008, 5.60001704202704564566319630582, 6.04568926384736410476689029063, 6.88889644550852025899291596638, 8.647701032249732197105625060814, 9.953219126276000935064581988431, 10.87318023524327909627572693642, 11.37590417769541719802945222772
