| L(s) = 1 | + (−3.20 − 4.41i)2-s + (−88.5 − 28.7i)3-s + (30.3 − 93.4i)4-s + (−278. − 28.1i)5-s + (156. + 482. i)6-s − 932. i·7-s + (−1.17e3 + 381. i)8-s + (5.23e3 + 3.80e3i)9-s + (767. + 1.31e3i)10-s + (−1.81e3 + 1.31e3i)11-s + (−5.37e3 + 7.39e3i)12-s + (4.77e3 − 6.56e3i)13-s + (−4.11e3 + 2.99e3i)14-s + (2.38e4 + 1.04e4i)15-s + (−4.72e3 − 3.42e3i)16-s + (−6.21e3 + 2.02e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.283 − 0.390i)2-s + (−1.89 − 0.614i)3-s + (0.237 − 0.729i)4-s + (−0.994 − 0.100i)5-s + (0.296 + 0.912i)6-s − 1.02i·7-s + (−0.810 + 0.263i)8-s + (2.39 + 1.73i)9-s + (0.242 + 0.416i)10-s + (−0.410 + 0.298i)11-s + (−0.897 + 1.23i)12-s + (0.602 − 0.828i)13-s + (−0.401 + 0.291i)14-s + (1.82 + 0.802i)15-s + (−0.288 − 0.209i)16-s + (−0.307 + 0.0997i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.380 - 0.924i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.380 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0605546 + 0.0405650i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0605546 + 0.0405650i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (278. + 28.1i)T \) |
| good | 2 | \( 1 + (3.20 + 4.41i)T + (-39.5 + 121. i)T^{2} \) |
| 3 | \( 1 + (88.5 + 28.7i)T + (1.76e3 + 1.28e3i)T^{2} \) |
| 7 | \( 1 + 932. iT - 8.23e5T^{2} \) |
| 11 | \( 1 + (1.81e3 - 1.31e3i)T + (6.02e6 - 1.85e7i)T^{2} \) |
| 13 | \( 1 + (-4.77e3 + 6.56e3i)T + (-1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (6.21e3 - 2.02e3i)T + (3.31e8 - 2.41e8i)T^{2} \) |
| 19 | \( 1 + (2.77e3 + 8.55e3i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 + (1.54e4 + 2.12e4i)T + (-1.05e9 + 3.23e9i)T^{2} \) |
| 29 | \( 1 + (922. - 2.83e3i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (1.29e4 + 3.98e4i)T + (-2.22e10 + 1.61e10i)T^{2} \) |
| 37 | \( 1 + (2.48e5 - 3.41e5i)T + (-2.93e10 - 9.02e10i)T^{2} \) |
| 41 | \( 1 + (-3.42e5 - 2.48e5i)T + (6.01e10 + 1.85e11i)T^{2} \) |
| 43 | \( 1 - 4.54e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + (-3.39e5 - 1.10e5i)T + (4.09e11 + 2.97e11i)T^{2} \) |
| 53 | \( 1 + (-2.89e5 - 9.40e4i)T + (9.50e11 + 6.90e11i)T^{2} \) |
| 59 | \( 1 + (1.36e6 + 9.91e5i)T + (7.69e11 + 2.36e12i)T^{2} \) |
| 61 | \( 1 + (2.75e5 - 2.00e5i)T + (9.71e11 - 2.98e12i)T^{2} \) |
| 67 | \( 1 + (2.57e6 - 8.36e5i)T + (4.90e12 - 3.56e12i)T^{2} \) |
| 71 | \( 1 + (-4.13e5 + 1.27e6i)T + (-7.35e12 - 5.34e12i)T^{2} \) |
| 73 | \( 1 + (2.95e6 + 4.06e6i)T + (-3.41e12 + 1.05e13i)T^{2} \) |
| 79 | \( 1 + (-2.01e6 + 6.20e6i)T + (-1.55e13 - 1.12e13i)T^{2} \) |
| 83 | \( 1 + (-2.75e6 + 8.94e5i)T + (2.19e13 - 1.59e13i)T^{2} \) |
| 89 | \( 1 + (6.04e6 - 4.39e6i)T + (1.36e13 - 4.20e13i)T^{2} \) |
| 97 | \( 1 + (-5.65e6 - 1.83e6i)T + (6.53e13 + 4.74e13i)T^{2} \) |
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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
−15.42302570729632193463701214385, −13.22254994898003316748855722025, −12.01199672240508739487309953913, −10.94996276550128216847790440859, −10.40400098541275802072493369373, −7.57373810785475983885357498752, −6.29332478251174665072394241151, −4.74054669159753464581300065959, −1.08501229829494968788372811797, −0.06471488770430665780552423669,
3.98989403599193957991120242349, 5.76275702386293964068408246385, 7.08009622824324626758961133395, 8.938016027378284928251966966102, 10.89898650387448998991134789875, 11.80845382694679412901206905465, 12.47635644647454628482366464045, 15.46661547747599075920012636076, 15.89953338274865565663276927968, 16.75911832211369234519357998788
