| L(s) = 1 | + (1.24 + 3.82i)2-s + (6.55 + 4.76i)3-s + (−6.59 + 4.79i)4-s + (2.37 − 7.31i)5-s + (−10.0 + 30.9i)6-s + (−5.66 + 4.11i)7-s + (−0.510 − 0.370i)8-s + (11.9 + 36.7i)9-s + 30.8·10-s + (−11.3 − 34.6i)11-s − 66.0·12-s + (−14.6 − 45.0i)13-s + (−22.7 − 16.5i)14-s + (50.3 − 36.6i)15-s + (−19.3 + 59.6i)16-s + (13.5 − 41.6i)17-s + ⋯ |
| L(s) = 1 | + (0.439 + 1.35i)2-s + (1.26 + 0.916i)3-s + (−0.824 + 0.599i)4-s + (0.212 − 0.653i)5-s + (−0.684 + 2.10i)6-s + (−0.305 + 0.222i)7-s + (−0.0225 − 0.0163i)8-s + (0.442 + 1.36i)9-s + 0.977·10-s + (−0.311 − 0.950i)11-s − 1.58·12-s + (−0.312 − 0.961i)13-s + (−0.434 − 0.315i)14-s + (0.867 − 0.630i)15-s + (−0.302 + 0.932i)16-s + (0.192 − 0.593i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.552 - 0.833i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.552 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.23714 + 2.30376i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23714 + 2.30376i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (5.66 - 4.11i)T \) |
| 11 | \( 1 + (11.3 + 34.6i)T \) |
| good | 2 | \( 1 + (-1.24 - 3.82i)T + (-6.47 + 4.70i)T^{2} \) |
| 3 | \( 1 + (-6.55 - 4.76i)T + (8.34 + 25.6i)T^{2} \) |
| 5 | \( 1 + (-2.37 + 7.31i)T + (-101. - 73.4i)T^{2} \) |
| 13 | \( 1 + (14.6 + 45.0i)T + (-1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-13.5 + 41.6i)T + (-3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (20.7 + 15.0i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + 1.91T + 1.21e4T^{2} \) |
| 29 | \( 1 + (151. - 110. i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-42.0 - 129. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-333. + 242. i)T + (1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-230. - 167. i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 + 525.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-222. - 161. i)T + (3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-11.3 - 34.8i)T + (-1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (569. - 413. i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-47.8 + 147. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + 712.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-121. + 374. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-211. + 153. i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (387. + 1.19e3i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (367. - 1.12e3i)T + (-4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 + 81.4T + 7.04e5T^{2} \) |
| 97 | \( 1 + (135. + 415. i)T + (-7.38e5 + 5.36e5i)T^{2} \) |
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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
−14.62755095636321429683608597416, −13.65342898131902174748716854093, −12.86870091482340951771715643144, −10.75082045334657632483012749096, −9.356753742322881673352935212387, −8.547954115052751867086165204696, −7.57830460670115684056709954933, −5.77714101407550611916643414320, −4.70811445946890095145610763780, −3.10329266390647691369051443883,
1.83035828783825441122800179490, 2.75375065247667869853547893484, 4.14582431470058849813809141317, 6.72377380687694936576716691861, 7.77249238727516886342263262612, 9.399776236702317319980875341289, 10.27673066900317883350270884797, 11.65400793277099556806744442919, 12.72617561342706494848128746935, 13.36187645531242399803206596937
