| L(s) = 1 | + 1.30·3-s + 15.1·5-s + 22.0·7-s − 25.3·9-s − 46.1·11-s + 57.4·13-s + 19.6·15-s − 52.7·17-s − 45.2·19-s + 28.7·21-s − 80.1·23-s + 103.·25-s − 68.0·27-s + 29·29-s + 47.9·31-s − 60.0·33-s + 333.·35-s + 91.2·37-s + 74.7·39-s + 252.·41-s + 480.·43-s − 382.·45-s + 551.·47-s + 145.·49-s − 68.6·51-s + 263.·53-s − 696.·55-s + ⋯ |
| L(s) = 1 | + 0.250·3-s + 1.35·5-s + 1.19·7-s − 0.937·9-s − 1.26·11-s + 1.22·13-s + 0.338·15-s − 0.751·17-s − 0.545·19-s + 0.298·21-s − 0.726·23-s + 0.824·25-s − 0.485·27-s + 0.185·29-s + 0.277·31-s − 0.316·33-s + 1.61·35-s + 0.405·37-s + 0.307·39-s + 0.961·41-s + 1.70·43-s − 1.26·45-s + 1.71·47-s + 0.423·49-s − 0.188·51-s + 0.682·53-s − 1.70·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.364444851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.364444851\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 - 1.30T + 27T^{2} \) |
| 5 | \( 1 - 15.1T + 125T^{2} \) |
| 7 | \( 1 - 22.0T + 343T^{2} \) |
| 11 | \( 1 + 46.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 57.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 52.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 45.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 80.1T + 1.21e4T^{2} \) |
| 31 | \( 1 - 47.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 91.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 252.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 480.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 551.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 263.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 315.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 628.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 693.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 785.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 807.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 299.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 333.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 34.5T + 7.04e5T^{2} \) |
| 97 | \( 1 - 929.T + 9.12e5T^{2} \) |
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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
−8.725861505160178386674185412029, −8.308330723447764235069393054806, −7.47491838072340068909039368439, −6.14730841246903259205452439339, −5.78483622760054238226346304546, −4.97471540727120655212253461720, −3.94460052817773575891922271071, −2.44743822183949838666169359284, −2.19113994726656099237517996666, −0.847655264579190982399869653147,
0.847655264579190982399869653147, 2.19113994726656099237517996666, 2.44743822183949838666169359284, 3.94460052817773575891922271071, 4.97471540727120655212253461720, 5.78483622760054238226346304546, 6.14730841246903259205452439339, 7.47491838072340068909039368439, 8.308330723447764235069393054806, 8.725861505160178386674185412029
