| L(s) = 1 | + (1.78 + 0.283i)2-s + (6.80 + 13.3i)3-s + (−57.7 − 18.7i)4-s + (−112. − 53.5i)5-s + (8.39 + 25.8i)6-s + (−120. − 120. i)7-s + (−201. − 102. i)8-s + (296. − 407. i)9-s + (−186. − 127. i)10-s + (−782. + 568. i)11-s + (−142. − 899. i)12-s + (−3.12e3 + 495. i)13-s + (−180. − 248. i)14-s + (−53.8 − 1.87e3i)15-s + (2.81e3 + 2.04e3i)16-s + (−11.6 + 22.8i)17-s + ⋯ |
| L(s) = 1 | + (0.223 + 0.0354i)2-s + (0.252 + 0.494i)3-s + (−0.902 − 0.293i)4-s + (−0.903 − 0.428i)5-s + (0.0388 + 0.119i)6-s + (−0.349 − 0.349i)7-s + (−0.392 − 0.200i)8-s + (0.406 − 0.559i)9-s + (−0.186 − 0.127i)10-s + (−0.587 + 0.426i)11-s + (−0.0824 − 0.520i)12-s + (−1.42 + 0.225i)13-s + (−0.0658 − 0.0905i)14-s + (−0.0159 − 0.555i)15-s + (0.686 + 0.499i)16-s + (−0.00237 + 0.00465i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.878 + 0.477i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.878 + 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0789782 - 0.311051i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0789782 - 0.311051i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (112. + 53.5i)T \) |
| good | 2 | \( 1 + (-1.78 - 0.283i)T + (60.8 + 19.7i)T^{2} \) |
| 3 | \( 1 + (-6.80 - 13.3i)T + (-428. + 589. i)T^{2} \) |
| 7 | \( 1 + (120. + 120. i)T + 1.17e5iT^{2} \) |
| 11 | \( 1 + (782. - 568. i)T + (5.47e5 - 1.68e6i)T^{2} \) |
| 13 | \( 1 + (3.12e3 - 495. i)T + (4.59e6 - 1.49e6i)T^{2} \) |
| 17 | \( 1 + (11.6 - 22.8i)T + (-1.41e7 - 1.95e7i)T^{2} \) |
| 19 | \( 1 + (-3.05e3 + 992. i)T + (3.80e7 - 2.76e7i)T^{2} \) |
| 23 | \( 1 + (528. - 3.33e3i)T + (-1.40e8 - 4.57e7i)T^{2} \) |
| 29 | \( 1 + (1.55e4 + 5.06e3i)T + (4.81e8 + 3.49e8i)T^{2} \) |
| 31 | \( 1 + (8.46e3 + 2.60e4i)T + (-7.18e8 + 5.21e8i)T^{2} \) |
| 37 | \( 1 + (1.06e4 + 6.75e4i)T + (-2.44e9 + 7.92e8i)T^{2} \) |
| 41 | \( 1 + (7.92e4 + 5.75e4i)T + (1.46e9 + 4.51e9i)T^{2} \) |
| 43 | \( 1 + (9.32e4 - 9.32e4i)T - 6.32e9iT^{2} \) |
| 47 | \( 1 + (-8.70e4 + 4.43e4i)T + (6.33e9 - 8.72e9i)T^{2} \) |
| 53 | \( 1 + (-3.89e4 - 7.63e4i)T + (-1.30e10 + 1.79e10i)T^{2} \) |
| 59 | \( 1 + (-1.20e5 + 1.66e5i)T + (-1.30e10 - 4.01e10i)T^{2} \) |
| 61 | \( 1 + (2.87e5 - 2.09e5i)T + (1.59e10 - 4.89e10i)T^{2} \) |
| 67 | \( 1 + (7.08e4 - 1.38e5i)T + (-5.31e10 - 7.31e10i)T^{2} \) |
| 71 | \( 1 + (-1.71e5 + 5.28e5i)T + (-1.03e11 - 7.52e10i)T^{2} \) |
| 73 | \( 1 + (1.84e4 - 1.16e5i)T + (-1.43e11 - 4.67e10i)T^{2} \) |
| 79 | \( 1 + (4.89e5 + 1.58e5i)T + (1.96e11 + 1.42e11i)T^{2} \) |
| 83 | \( 1 + (2.05e5 + 1.04e5i)T + (1.92e11 + 2.64e11i)T^{2} \) |
| 89 | \( 1 + (-6.01e5 - 8.27e5i)T + (-1.53e11 + 4.72e11i)T^{2} \) |
| 97 | \( 1 + (3.05e5 - 1.55e5i)T + (4.89e11 - 6.73e11i)T^{2} \) |
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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
−15.48375773464899919058816827399, −14.71014672758360584290619254314, −13.16117103752774305565509845314, −12.11301154100750895680082920302, −10.09533116710283925384442426642, −9.137800176963035913939539606359, −7.42568048149829525120833247013, −4.97079027124304707447900268490, −3.78768288761012766174080104075, −0.17082271159831108725425217378,
3.04255072752865035292978004546, 4.93660270875369824253696840036, 7.31219471895492282548158884773, 8.404749037598114118733638755438, 10.15753511754212211210690782015, 12.00971728371829440745204777466, 12.95839222400784070993916256241, 14.15597465416233285944571408512, 15.39305708472863307545499749475, 16.78394308817044721761596966397
