| L(s) = 1 | + (−1.35 + 5.01i)3-s + (3.33 − 5.78i)5-s + (14.9 + 10.8i)7-s + (−23.3 − 13.6i)9-s + (31.4 − 18.1i)11-s + (39.1 − 22.5i)13-s + (24.4 + 24.5i)15-s + (−39.7 + 68.7i)17-s + (80.2 − 46.3i)19-s + (−74.9 + 60.4i)21-s + (−12.2 − 7.08i)23-s + (40.2 + 69.6i)25-s + (99.9 − 98.4i)27-s + (9.72 + 5.61i)29-s + 124. i·31-s + ⋯ |
| L(s) = 1 | + (−0.261 + 0.965i)3-s + (0.298 − 0.517i)5-s + (0.809 + 0.587i)7-s + (−0.863 − 0.504i)9-s + (0.861 − 0.497i)11-s + (0.835 − 0.482i)13-s + (0.421 + 0.423i)15-s + (−0.566 + 0.981i)17-s + (0.968 − 0.559i)19-s + (−0.778 + 0.627i)21-s + (−0.111 − 0.0642i)23-s + (0.321 + 0.557i)25-s + (0.712 − 0.702i)27-s + (0.0622 + 0.0359i)29-s + 0.719i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 - 0.777i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.981596567\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.981596567\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.35 - 5.01i)T \) |
| 7 | \( 1 + (-14.9 - 10.8i)T \) |
| good | 5 | \( 1 + (-3.33 + 5.78i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-31.4 + 18.1i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-39.1 + 22.5i)T + (1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (39.7 - 68.7i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-80.2 + 46.3i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (12.2 + 7.08i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-9.72 - 5.61i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 124. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (-92.0 - 159. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-117. - 203. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (157. - 272. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 - 48.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + (5.19 + 2.99i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 - 60.6T + 2.05e5T^{2} \) |
| 61 | \( 1 + 381. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 696.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 831. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-505. - 292. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + 514.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-138. + 239. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (807. + 1.39e3i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (177. + 102. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−11.41098250922792288238711566461, −11.01437445498129332056384665150, −9.716573980045619240144889614763, −8.857229324599614081707624515847, −8.249290592406542334964754057864, −6.37391829427595910734808585632, −5.47233157841275548671725506397, −4.52948937143623575086204299636, −3.23907635064527876008726418694, −1.26619036233168417683090371381,
1.03208486567949616737761459697, 2.26028224474411521433342178940, 4.02940276656099055265143006117, 5.46466054370668729826054394540, 6.66343308425666900394913918565, 7.27659406010604348747445589213, 8.369639146306460026236208975308, 9.525338717326746730372148781459, 10.80592077888066097772852800732, 11.49835249359742244740984725091
