| L(s) = 1 | + (−11.5 + 19.9i)3-s + (208. + 361. i)5-s + (9.57e3 + 1.65e4i)9-s + (1.30e3 − 2.25e3i)11-s − 1.33e4·13-s − 9.63e3·15-s + (2.43e5 − 4.21e5i)17-s + (2.92e5 + 5.07e5i)19-s + (5.89e5 + 1.02e6i)23-s + (8.89e5 − 1.54e6i)25-s − 8.95e5·27-s − 1.60e6·29-s + (−1.32e6 + 2.29e6i)31-s + (3.00e4 + 5.20e4i)33-s + (−5.03e6 − 8.71e6i)37-s + ⋯ |
| L(s) = 1 | + (−0.0822 + 0.142i)3-s + (0.149 + 0.258i)5-s + (0.486 + 0.842i)9-s + (0.0268 − 0.0464i)11-s − 0.129·13-s − 0.0491·15-s + (0.707 − 1.22i)17-s + (0.515 + 0.893i)19-s + (0.438 + 0.760i)23-s + (0.455 − 0.788i)25-s − 0.324·27-s − 0.422·29-s + (−0.257 + 0.445i)31-s + (0.00440 + 0.00763i)33-s + (−0.441 − 0.764i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.267234957\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.267234957\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (11.5 - 19.9i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-208. - 361. i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 11 | \( 1 + (-1.30e3 + 2.25e3i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + 1.33e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + (-2.43e5 + 4.21e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-2.92e5 - 5.07e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-5.89e5 - 1.02e6i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + 1.60e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (1.32e6 - 2.29e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (5.03e6 + 8.71e6i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 - 1.51e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.51e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (4.25e6 + 7.37e6i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (2.45e7 - 4.25e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-5.06e7 + 8.76e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-2.99e7 - 5.19e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-7.96e7 + 1.37e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + 2.27e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + (1.09e8 - 1.90e8i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-1.18e8 - 2.04e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 - 2.85e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (5.86e7 + 1.01e8i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 - 1.59e9T + 7.60e17T^{2} \) |
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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
−10.92787748427134570800415489133, −10.07881096189137204558011266142, −9.215222967077307792794931679892, −7.82280431692287734111287804353, −7.14176985885183428107055354231, −5.72238902244702908028098388745, −4.83707714423310427731038191137, −3.49701434356348388230896587312, −2.26917140674021083960640413364, −0.980433458620367159653351949718,
0.60150159358588029568963239834, 1.58964102439517771150108954291, 3.11335781302291420713994472009, 4.27024776669537874689395854072, 5.50375233218565424092668372580, 6.57429465773339870955770391253, 7.54610510480449737851536954170, 8.788343305727316992021565276677, 9.603233164500168253435056961709, 10.63069376737815925263956616350
