| L(s) = 1 | + 2-s − 3.00·3-s + 4-s − 0.777·5-s − 3.00·6-s + 3.31·7-s + 8-s + 6.00·9-s − 0.777·10-s + 3.00·11-s − 3.00·12-s + 0.139·13-s + 3.31·14-s + 2.33·15-s + 16-s + 6.00·17-s + 6.00·18-s + 0.993·19-s − 0.777·20-s − 9.94·21-s + 3.00·22-s + 7.61·23-s − 3.00·24-s − 4.39·25-s + 0.139·26-s − 9.01·27-s + 3.31·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 0.5·4-s − 0.347·5-s − 1.22·6-s + 1.25·7-s + 0.353·8-s + 2.00·9-s − 0.245·10-s + 0.907·11-s − 0.866·12-s + 0.0387·13-s + 0.886·14-s + 0.602·15-s + 0.250·16-s + 1.45·17-s + 1.41·18-s + 0.227·19-s − 0.173·20-s − 2.17·21-s + 0.641·22-s + 1.58·23-s − 0.612·24-s − 0.879·25-s + 0.0274·26-s − 1.73·27-s + 0.626·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6038 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6038 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.470284097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.470284097\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3019 | \( 1+O(T) \) |
| good | 3 | \( 1 + 3.00T + 3T^{2} \) |
| 5 | \( 1 + 0.777T + 5T^{2} \) |
| 7 | \( 1 - 3.31T + 7T^{2} \) |
| 11 | \( 1 - 3.00T + 11T^{2} \) |
| 13 | \( 1 - 0.139T + 13T^{2} \) |
| 17 | \( 1 - 6.00T + 17T^{2} \) |
| 19 | \( 1 - 0.993T + 19T^{2} \) |
| 23 | \( 1 - 7.61T + 23T^{2} \) |
| 29 | \( 1 - 4.89T + 29T^{2} \) |
| 31 | \( 1 - 1.91T + 31T^{2} \) |
| 37 | \( 1 - 0.677T + 37T^{2} \) |
| 41 | \( 1 - 5.92T + 41T^{2} \) |
| 43 | \( 1 - 1.00T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 3.06T + 53T^{2} \) |
| 59 | \( 1 + 2.62T + 59T^{2} \) |
| 61 | \( 1 - 8.78T + 61T^{2} \) |
| 67 | \( 1 + 14.8T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 + 0.642T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−7.68882988308615720502432035553, −7.24684717915289062027626530992, −6.48744199792220512715372184704, −5.64811554795846840669226666371, −5.38297893829833551958926699290, −4.46151607997470056629052339850, −4.14375534960916113827582540748, −2.93183811368706048475644154896, −1.44492520329279356322069733960, −0.953660258155975150306142667219,
0.953660258155975150306142667219, 1.44492520329279356322069733960, 2.93183811368706048475644154896, 4.14375534960916113827582540748, 4.46151607997470056629052339850, 5.38297893829833551958926699290, 5.64811554795846840669226666371, 6.48744199792220512715372184704, 7.24684717915289062027626530992, 7.68882988308615720502432035553
