| L(s) = 1 | + (−2.66 + 3.92i)2-s + (−2.04 + 4.77i)3-s + (−5.37 − 13.4i)4-s + (−8.08 + 17.4i)5-s + (−13.3 − 20.7i)6-s + (−2.26 − 8.16i)7-s + (30.1 + 6.64i)8-s + (−18.6 − 19.5i)9-s + (−47.0 − 78.2i)10-s + (1.34 + 1.27i)11-s + (75.3 + 1.84i)12-s + (−3.48 + 32.0i)13-s + (38.1 + 12.8i)14-s + (−66.9 − 74.2i)15-s + (−22.1 + 20.9i)16-s + (−51.4 − 14.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.941 + 1.38i)2-s + (−0.392 + 0.919i)3-s + (−0.671 − 1.68i)4-s + (−0.723 + 1.56i)5-s + (−0.907 − 1.41i)6-s + (−0.122 − 0.441i)7-s + (1.33 + 0.293i)8-s + (−0.691 − 0.722i)9-s + (−1.48 − 2.47i)10-s + (0.0367 + 0.0348i)11-s + (1.81 + 0.0442i)12-s + (−0.0744 + 0.684i)13-s + (0.727 + 0.245i)14-s + (−1.15 − 1.27i)15-s + (−0.345 + 0.327i)16-s + (−0.734 − 0.203i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00404i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0677872 + 0.000137060i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0677872 + 0.000137060i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.04 - 4.77i)T \) |
| 59 | \( 1 + (-192. + 410. i)T \) |
| good | 2 | \( 1 + (2.66 - 3.92i)T + (-2.96 - 7.43i)T^{2} \) |
| 5 | \( 1 + (8.08 - 17.4i)T + (-80.9 - 95.2i)T^{2} \) |
| 7 | \( 1 + (2.26 + 8.16i)T + (-293. + 176. i)T^{2} \) |
| 11 | \( 1 + (-1.34 - 1.27i)T + (72.0 + 1.32e3i)T^{2} \) |
| 13 | \( 1 + (3.48 - 32.0i)T + (-2.14e3 - 472. i)T^{2} \) |
| 17 | \( 1 + (51.4 + 14.2i)T + (4.20e3 + 2.53e3i)T^{2} \) |
| 19 | \( 1 + (28.1 + 21.3i)T + (1.83e3 + 6.60e3i)T^{2} \) |
| 23 | \( 1 + (-46.3 - 87.5i)T + (-6.82e3 + 1.00e4i)T^{2} \) |
| 29 | \( 1 + (52.8 - 35.8i)T + (9.02e3 - 2.26e4i)T^{2} \) |
| 31 | \( 1 + (-99.0 - 130. i)T + (-7.96e3 + 2.87e4i)T^{2} \) |
| 37 | \( 1 + (-71.9 - 326. i)T + (-4.59e4 + 2.12e4i)T^{2} \) |
| 41 | \( 1 + (229. + 121. i)T + (3.86e4 + 5.70e4i)T^{2} \) |
| 43 | \( 1 + (-231. - 244. i)T + (-4.30e3 + 7.93e4i)T^{2} \) |
| 47 | \( 1 + (424. - 196. i)T + (6.72e4 - 7.91e4i)T^{2} \) |
| 53 | \( 1 + (-334. + 556. i)T + (-6.97e4 - 1.31e5i)T^{2} \) |
| 61 | \( 1 + (609. + 413. i)T + (8.40e4 + 2.10e5i)T^{2} \) |
| 67 | \( 1 + (-112. + 512. i)T + (-2.72e5 - 1.26e5i)T^{2} \) |
| 71 | \( 1 + (411. + 889. i)T + (-2.31e5 + 2.72e5i)T^{2} \) |
| 73 | \( 1 + (-80.5 + 239. i)T + (-3.09e5 - 2.35e5i)T^{2} \) |
| 79 | \( 1 + (60.6 - 1.11e3i)T + (-4.90e5 - 5.33e4i)T^{2} \) |
| 83 | \( 1 + (-195. + 1.19e3i)T + (-5.41e5 - 1.82e5i)T^{2} \) |
| 89 | \( 1 + (-564. - 832. i)T + (-2.60e5 + 6.54e5i)T^{2} \) |
| 97 | \( 1 + (197. + 586. i)T + (-7.26e5 + 5.52e5i)T^{2} \) |
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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
−13.80399899982544652047924198052, −11.70743992154833428311200607347, −10.91362052141891514008001220060, −10.09189210475597978578742394079, −9.180181751968318923863570686296, −7.984024393966654106106276230819, −6.83889811134358330755484621501, −6.44379743750622362060946191735, −4.80610536029879032061838157046, −3.36181883466073882875543291244,
0.05463713038442802338801139884, 1.04127545035761684929089712084, 2.43264638309204145461983619957, 4.24034054138083554258918908705, 5.71943312478006385324170993920, 7.58435248210307400326584761556, 8.525378235311133928730985129484, 8.977915338530277044669469143347, 10.42597799966744592677824503534, 11.46078353818616592719332973590
