| L(s) = 1 | + (−0.570 + 0.0299i)2-s + (1.41 − 1.74i)3-s + (−7.63 + 0.802i)4-s + (−10.8 + 2.77i)5-s + (−0.756 + 1.04i)6-s + (18.4 + 0.955i)7-s + (8.84 − 1.40i)8-s + (4.55 + 21.4i)9-s + (6.09 − 1.90i)10-s + (−42.9 − 9.13i)11-s + (−9.41 + 14.4i)12-s + (63.7 − 32.4i)13-s + (−10.5 + 0.00760i)14-s + (−10.4 + 22.8i)15-s + (55.0 − 11.6i)16-s + (44.2 − 115. i)17-s + ⋯ |
| L(s) = 1 | + (−0.201 + 0.0105i)2-s + (0.272 − 0.336i)3-s + (−0.953 + 0.100i)4-s + (−0.968 + 0.248i)5-s + (−0.0514 + 0.0708i)6-s + (0.998 + 0.0516i)7-s + (0.390 − 0.0619i)8-s + (0.168 + 0.794i)9-s + (0.192 − 0.0603i)10-s + (−1.17 − 0.250i)11-s + (−0.226 + 0.348i)12-s + (1.35 − 0.692i)13-s + (−0.202 + 0.000145i)14-s + (−0.180 + 0.393i)15-s + (0.860 − 0.182i)16-s + (0.631 − 1.64i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 + 0.660i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.750 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.11841 - 0.421948i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11841 - 0.421948i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (10.8 - 2.77i)T \) |
| 7 | \( 1 + (-18.4 - 0.955i)T \) |
| good | 2 | \( 1 + (0.570 - 0.0299i)T + (7.95 - 0.836i)T^{2} \) |
| 3 | \( 1 + (-1.41 + 1.74i)T + (-5.61 - 26.4i)T^{2} \) |
| 11 | \( 1 + (42.9 + 9.13i)T + (1.21e3 + 541. i)T^{2} \) |
| 13 | \( 1 + (-63.7 + 32.4i)T + (1.29e3 - 1.77e3i)T^{2} \) |
| 17 | \( 1 + (-44.2 + 115. i)T + (-3.65e3 - 3.28e3i)T^{2} \) |
| 19 | \( 1 + (2.03 - 19.3i)T + (-6.70e3 - 1.42e3i)T^{2} \) |
| 23 | \( 1 + (5.56 + 106. i)T + (-1.21e4 + 1.27e3i)T^{2} \) |
| 29 | \( 1 + (0.465 + 0.640i)T + (-7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-2.69 - 6.05i)T + (-1.99e4 + 2.21e4i)T^{2} \) |
| 37 | \( 1 + (-305. - 198. i)T + (2.06e4 + 4.62e4i)T^{2} \) |
| 41 | \( 1 + (113. + 36.7i)T + (5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + (-18.9 + 18.9i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + (-525. + 201. i)T + (7.71e4 - 6.94e4i)T^{2} \) |
| 53 | \( 1 + (109. + 89.0i)T + (3.09e4 + 1.45e5i)T^{2} \) |
| 59 | \( 1 + (-499. - 554. i)T + (-2.14e4 + 2.04e5i)T^{2} \) |
| 61 | \( 1 + (430. + 387. i)T + (2.37e4 + 2.25e5i)T^{2} \) |
| 67 | \( 1 + (389. + 149. i)T + (2.23e5 + 2.01e5i)T^{2} \) |
| 71 | \( 1 + (-107. + 78.3i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (245. + 377. i)T + (-1.58e5 + 3.55e5i)T^{2} \) |
| 79 | \( 1 + (50.0 - 112. i)T + (-3.29e5 - 3.66e5i)T^{2} \) |
| 83 | \( 1 + (23.6 + 149. i)T + (-5.43e5 + 1.76e5i)T^{2} \) |
| 89 | \( 1 + (-393. + 437. i)T + (-7.36e4 - 7.01e5i)T^{2} \) |
| 97 | \( 1 + (221. - 1.39e3i)T + (-8.68e5 - 2.82e5i)T^{2} \) |
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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
−12.18170316688670297223202706026, −11.01547786940007034672067362815, −10.32881498224520909980291220392, −8.665177168546088139732537860791, −8.043379824662921419657158937483, −7.48712554445071272019903792777, −5.39752399278519783236557611689, −4.45992177072927893520872812543, −2.93257870759969390599458755534, −0.76106737215995713429228777714,
1.15768507250732171109779062139, 3.72553138361596334432583966665, 4.39247305426852343218582258434, 5.73735038490609356170994001591, 7.64482271515711812814414700831, 8.360789972555113442430544432695, 9.115684148002768865350421167145, 10.38399044284884828825527566878, 11.24479204952444672362364110701, 12.44315280858663289085522143865
