| L(s) = 1 | + (−0.5 + 0.866i)2-s + (2 − 3.46i)3-s + (3.5 + 6.06i)4-s + (−12.9 − 7.49i)5-s + (1.99 + 3.46i)6-s + (−31.4 − 18.1i)7-s − 15·8-s + (5.50 + 9.52i)9-s + (12.9 − 7.49i)10-s + (1.48 + 2.56i)11-s + 28·12-s + (−19.9 − 11.4i)13-s + (31.4 − 18.1i)14-s + (−51.9 + 29.9i)15-s + (−20.5 + 35.5i)16-s + (−15.3 + 8.86i)17-s + ⋯ |
| L(s) = 1 | + (−0.176 + 0.306i)2-s + (0.384 − 0.666i)3-s + (0.437 + 0.757i)4-s + (−1.16 − 0.670i)5-s + (0.136 + 0.235i)6-s + (−1.69 − 0.980i)7-s − 0.662·8-s + (0.203 + 0.352i)9-s + (0.410 − 0.237i)10-s + (0.0406 + 0.0703i)11-s + 0.673·12-s + (−0.424 − 0.245i)13-s + (0.600 − 0.346i)14-s + (−0.893 + 0.516i)15-s + (−0.320 + 0.554i)16-s + (−0.219 + 0.126i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 + 0.514i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.108776 - 0.392764i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.108776 - 0.392764i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 97 | \( 1 + (719. - 628. i)T \) |
| good | 2 | \( 1 + (0.5 - 0.866i)T + (-4 - 6.92i)T^{2} \) |
| 3 | \( 1 + (-2 + 3.46i)T + (-13.5 - 23.3i)T^{2} \) |
| 5 | \( 1 + (12.9 + 7.49i)T + (62.5 + 108. i)T^{2} \) |
| 7 | \( 1 + (31.4 + 18.1i)T + (171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-1.48 - 2.56i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (19.9 + 11.4i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (15.3 - 8.86i)T + (2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + 143. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (93.9 - 54.2i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-82.9 - 47.8i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-40.8 + 70.8i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (266. + 153. i)T + (2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-377. - 217. i)T + (3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (47.6 - 82.5i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + 133.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (10.9 - 18.8i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (332. + 192. i)T + (1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-313. + 543. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + 615. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + (251. - 145. i)T + (1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-66.3 + 114. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 - 295.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (1.18e3 - 683. i)T + (2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + 788.T + 7.04e5T^{2} \) |
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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
−12.84164406877441548621072970913, −12.37634156241358759881529875742, −11.02580829763360026369107533416, −9.485719165672540435552603919642, −8.212009404103192855133052338119, −7.36149419919038059527831750751, −6.68571852687021556530162532566, −4.24289478565814443840141834621, −2.96143242919997982961260594468, −0.22105559565795520987164770936,
2.78641619210688083383868805691, 3.82797749190650081617841823326, 5.97801006550602130083322253461, 6.94352708294937318167664334821, 8.708238242297786472733451100058, 9.827120635942042650621240146158, 10.35429359108067609636462961906, 11.90363757397356597573466304799, 12.34671518910676885400280203844, 14.29060299886249015095509063733
