| L(s) = 1 | + (−3.16 − 1.82i)2-s + (−3.78 − 3.55i)3-s + (2.66 + 4.60i)4-s − 12.2·5-s + (5.46 + 18.1i)6-s + (4.78 − 17.8i)7-s + 9.77i·8-s + (1.65 + 26.9i)9-s + (38.6 + 22.3i)10-s − 10.0i·11-s + (6.33 − 26.9i)12-s + (57.0 + 32.9i)13-s + (−47.7 + 47.8i)14-s + (46.3 + 43.5i)15-s + (39.1 − 67.7i)16-s + (−38.6 + 66.9i)17-s + ⋯ |
| L(s) = 1 | + (−1.11 − 0.645i)2-s + (−0.728 − 0.685i)3-s + (0.332 + 0.576i)4-s − 1.09·5-s + (0.372 + 1.23i)6-s + (0.258 − 0.965i)7-s + 0.432i·8-s + (0.0614 + 0.998i)9-s + (1.22 + 0.705i)10-s − 0.274i·11-s + (0.152 − 0.647i)12-s + (1.21 + 0.702i)13-s + (−0.912 + 0.912i)14-s + (0.796 + 0.749i)15-s + (0.611 − 1.05i)16-s + (−0.551 + 0.955i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.409 - 0.912i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.409 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0905390 + 0.0585736i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0905390 + 0.0585736i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (3.78 + 3.55i)T \) |
| 7 | \( 1 + (-4.78 + 17.8i)T \) |
| good | 2 | \( 1 + (3.16 + 1.82i)T + (4 + 6.92i)T^{2} \) |
| 5 | \( 1 + 12.2T + 125T^{2} \) |
| 11 | \( 1 + 10.0iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (-57.0 - 32.9i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (38.6 - 66.9i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (104. - 60.1i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + 122. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (50.3 - 29.0i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (174. - 100. i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-82.9 - 143. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (112. - 194. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-152. - 263. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-11.2 + 19.5i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-360. - 207. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-180. - 312. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (635. + 367. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-55.6 - 96.3i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 455. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (662. + 382. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-21.5 + 37.3i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (200. + 346. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-347. - 602. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (955. - 551. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−14.63071932991150502460448144131, −13.25307763809079288633288115503, −12.01084672269609947138541870717, −10.93361749698252605812707391533, −10.66668230046742848472365782765, −8.605847716874923412011108085821, −7.84788010097121942725863081114, −6.41478606553514901971464115412, −4.20093058749318497014475024253, −1.45966320606046312034251609830,
0.12477247859863802823322876740, 3.95325893069035038081705911491, 5.72392084832478489633625181227, 7.18585726230135440291447157916, 8.511438405261400504192331577018, 9.304109423619973428440971007707, 10.81108087811668728363734014987, 11.62944011657408001537934873418, 12.90151666784622508234741544566, 15.26490069373250768917787289443
