| L(s) = 1 | + (−2.99 − 0.325i)2-s + (−1.94 + 2.28i)3-s + (1.05 + 0.233i)4-s + (−3.11 − 2.36i)5-s + (6.56 − 6.21i)6-s + (−9.33 + 23.4i)7-s + (19.7 + 6.65i)8-s + (−1.45 − 8.88i)9-s + (8.55 + 8.10i)10-s + (−16.8 + 7.78i)11-s + (−2.59 + 1.96i)12-s + (−5.52 + 33.6i)13-s + (35.6 − 67.1i)14-s + (11.4 − 2.52i)15-s + (−64.8 − 30.0i)16-s + (39.3 + 98.8i)17-s + ⋯ |
| L(s) = 1 | + (−1.05 − 0.115i)2-s + (−0.373 + 0.440i)3-s + (0.132 + 0.0291i)4-s + (−0.278 − 0.211i)5-s + (0.446 − 0.423i)6-s + (−0.504 + 1.26i)7-s + (0.872 + 0.294i)8-s + (−0.0539 − 0.328i)9-s + (0.270 + 0.256i)10-s + (−0.461 + 0.213i)11-s + (−0.0623 + 0.0473i)12-s + (−0.117 + 0.718i)13-s + (0.680 − 1.28i)14-s + (0.197 − 0.0434i)15-s + (−1.01 − 0.469i)16-s + (0.561 + 1.41i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 + 0.971i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0905009 - 0.115077i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0905009 - 0.115077i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.94 - 2.28i)T \) |
| 59 | \( 1 + (-324. - 316. i)T \) |
| good | 2 | \( 1 + (2.99 + 0.325i)T + (7.81 + 1.71i)T^{2} \) |
| 5 | \( 1 + (3.11 + 2.36i)T + (33.4 + 120. i)T^{2} \) |
| 7 | \( 1 + (9.33 - 23.4i)T + (-249. - 235. i)T^{2} \) |
| 11 | \( 1 + (16.8 - 7.78i)T + (861. - 1.01e3i)T^{2} \) |
| 13 | \( 1 + (5.52 - 33.6i)T + (-2.08e3 - 701. i)T^{2} \) |
| 17 | \( 1 + (-39.3 - 98.8i)T + (-3.56e3 + 3.37e3i)T^{2} \) |
| 19 | \( 1 + (60.0 + 88.6i)T + (-2.53e3 + 6.37e3i)T^{2} \) |
| 23 | \( 1 + (-2.56 + 47.3i)T + (-1.20e4 - 1.31e3i)T^{2} \) |
| 29 | \( 1 + (-74.6 + 8.11i)T + (2.38e4 - 5.24e3i)T^{2} \) |
| 31 | \( 1 + (-109. + 161. i)T + (-1.10e4 - 2.76e4i)T^{2} \) |
| 37 | \( 1 + (281. - 94.8i)T + (4.03e4 - 3.06e4i)T^{2} \) |
| 41 | \( 1 + (24.6 + 455. i)T + (-6.85e4 + 7.45e3i)T^{2} \) |
| 43 | \( 1 + (425. + 196. i)T + (5.14e4 + 6.05e4i)T^{2} \) |
| 47 | \( 1 + (-453. + 344. i)T + (2.77e4 - 1.00e5i)T^{2} \) |
| 53 | \( 1 + (312. - 296. i)T + (8.06e3 - 1.48e5i)T^{2} \) |
| 61 | \( 1 + (308. + 33.5i)T + (2.21e5 + 4.87e4i)T^{2} \) |
| 67 | \( 1 + (-907. - 305. i)T + (2.39e5 + 1.82e5i)T^{2} \) |
| 71 | \( 1 + (-238. + 181. i)T + (9.57e4 - 3.44e5i)T^{2} \) |
| 73 | \( 1 + (180. - 340. i)T + (-2.18e5 - 3.21e5i)T^{2} \) |
| 79 | \( 1 + (-132. - 155. i)T + (-7.97e4 + 4.86e5i)T^{2} \) |
| 83 | \( 1 + (-364. + 219. i)T + (2.67e5 - 5.05e5i)T^{2} \) |
| 89 | \( 1 + (1.54e3 - 168. i)T + (6.88e5 - 1.51e5i)T^{2} \) |
| 97 | \( 1 + (545. + 1.02e3i)T + (-5.12e5 + 7.55e5i)T^{2} \) |
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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
−11.85797089930416550319881618833, −10.62602628300880075265588575539, −9.934359567801418364798192219694, −8.827793641408855229687410095676, −8.382078674673371962711657101025, −6.77016294605473126599087468337, −5.47003196381334030502952048731, −4.23169526213510000837645941446, −2.22193189667351936058340722654, −0.12070293850768343271146312202,
1.08290220784700640475643945536, 3.40771835860283382287391635826, 4.99169546552644355686299156664, 6.66566192718560930735810173618, 7.53611484430463721651775918190, 8.190755221183676855456281209339, 9.712279656092618178598747269370, 10.33373897890346310250705886526, 11.20727540988899488637370559381, 12.54307240148976799175854163047
