| L(s) = 1 | + (−5.03 + 2.57i)2-s + (3.80 − 19.1i)3-s + (18.7 − 25.9i)4-s + (8.73 + 13.0i)5-s + (30.1 + 106. i)6-s + (126. + 84.6i)7-s + (−27.4 + 178. i)8-s + (−127. − 52.8i)9-s + (−77.6 − 43.3i)10-s + (307. − 61.2i)11-s + (−425. − 457. i)12-s + (−241. − 241. i)13-s + (−856. − 99.9i)14-s + (283. − 117. i)15-s + (−322. − 971. i)16-s + (1.04e3 − 570. i)17-s + ⋯ |
| L(s) = 1 | + (−0.890 + 0.455i)2-s + (0.244 − 1.22i)3-s + (0.585 − 0.810i)4-s + (0.156 + 0.233i)5-s + (0.341 + 1.20i)6-s + (0.977 + 0.652i)7-s + (−0.151 + 0.988i)8-s + (−0.524 − 0.217i)9-s + (−0.245 − 0.136i)10-s + (0.766 − 0.152i)11-s + (−0.852 − 0.916i)12-s + (−0.395 − 0.395i)13-s + (−1.16 − 0.136i)14-s + (0.325 − 0.134i)15-s + (−0.315 − 0.949i)16-s + (0.878 − 0.478i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 + 0.663i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.748 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.39275 - 0.528386i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39275 - 0.528386i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (5.03 - 2.57i)T \) |
| 17 | \( 1 + (-1.04e3 + 570. i)T \) |
| good | 3 | \( 1 + (-3.80 + 19.1i)T + (-224. - 92.9i)T^{2} \) |
| 5 | \( 1 + (-8.73 - 13.0i)T + (-1.19e3 + 2.88e3i)T^{2} \) |
| 7 | \( 1 + (-126. - 84.6i)T + (6.43e3 + 1.55e4i)T^{2} \) |
| 11 | \( 1 + (-307. + 61.2i)T + (1.48e5 - 6.16e4i)T^{2} \) |
| 13 | \( 1 + (241. + 241. i)T + 3.71e5iT^{2} \) |
| 19 | \( 1 + (-208. - 502. i)T + (-1.75e6 + 1.75e6i)T^{2} \) |
| 23 | \( 1 + (-185. - 932. i)T + (-5.94e6 + 2.46e6i)T^{2} \) |
| 29 | \( 1 + (-4.15e3 + 2.77e3i)T + (7.84e6 - 1.89e7i)T^{2} \) |
| 31 | \( 1 + (3.72e3 + 741. i)T + (2.64e7 + 1.09e7i)T^{2} \) |
| 37 | \( 1 + (9.47e3 + 1.88e3i)T + (6.40e7 + 2.65e7i)T^{2} \) |
| 41 | \( 1 + (-1.37e3 + 2.06e3i)T + (-4.43e7 - 1.07e8i)T^{2} \) |
| 43 | \( 1 + (-1.78e3 + 4.30e3i)T + (-1.03e8 - 1.03e8i)T^{2} \) |
| 47 | \( 1 + (3.56e3 - 3.56e3i)T - 2.29e8iT^{2} \) |
| 53 | \( 1 + (3.71e3 - 1.53e3i)T + (2.95e8 - 2.95e8i)T^{2} \) |
| 59 | \( 1 + (-4.71e4 - 1.95e4i)T + (5.05e8 + 5.05e8i)T^{2} \) |
| 61 | \( 1 + (2.67e4 + 1.78e4i)T + (3.23e8 + 7.80e8i)T^{2} \) |
| 67 | \( 1 - 6.16e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (1.44e4 - 7.27e4i)T + (-1.66e9 - 6.90e8i)T^{2} \) |
| 73 | \( 1 + (-1.77e4 - 2.65e4i)T + (-7.93e8 + 1.91e9i)T^{2} \) |
| 79 | \( 1 + (8.19e3 - 1.62e3i)T + (2.84e9 - 1.17e9i)T^{2} \) |
| 83 | \( 1 + (5.67e4 - 2.35e4i)T + (2.78e9 - 2.78e9i)T^{2} \) |
| 89 | \( 1 + (-5.69e4 + 5.69e4i)T - 5.58e9iT^{2} \) |
| 97 | \( 1 + (1.44e5 - 9.68e4i)T + (3.28e9 - 7.93e9i)T^{2} \) |
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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
−14.05173837990018867871731678554, −12.33766393112519009630104613287, −11.52481110069430171161301401367, −10.05064652924345877210108532861, −8.643019859466549192650195903508, −7.79003004062887357020724998195, −6.76456601536783031060903472191, −5.45420409840489497255398464154, −2.32469142724728196670234576412, −1.07529387188777331214697020372,
1.40253499186842185114316226947, 3.54713622108512946845812300253, 4.77959817137982079607725714850, 7.08778307867587432334169999973, 8.483318265959841107615724846986, 9.447568734546507013545163228167, 10.37577021824603122644679684087, 11.26292788400780255777466230978, 12.48131323376117681433699976412, 14.19195695187540250238017933506
