| L(s) = 1 | + (−4.61 − 0.241i)2-s + (1.85 + 2.29i)3-s + (13.3 + 1.39i)4-s + (−10.3 + 4.28i)5-s + (−8.00 − 11.0i)6-s + (−16.5 + 8.39i)7-s + (−24.5 − 3.88i)8-s + (3.80 − 17.9i)9-s + (48.7 − 17.2i)10-s + (12.7 − 2.71i)11-s + (21.4 + 33.0i)12-s + (−54.2 − 27.6i)13-s + (78.2 − 34.7i)14-s + (−28.9 − 15.7i)15-s + (7.70 + 1.63i)16-s + (27.3 + 71.3i)17-s + ⋯ |
| L(s) = 1 | + (−1.63 − 0.0855i)2-s + (0.356 + 0.440i)3-s + (1.66 + 0.174i)4-s + (−0.923 + 0.382i)5-s + (−0.544 − 0.749i)6-s + (−0.891 + 0.453i)7-s + (−1.08 − 0.171i)8-s + (0.141 − 0.663i)9-s + (1.54 − 0.545i)10-s + (0.350 − 0.0744i)11-s + (0.516 + 0.795i)12-s + (−1.15 − 0.589i)13-s + (1.49 − 0.663i)14-s + (−0.498 − 0.270i)15-s + (0.120 + 0.0255i)16-s + (0.390 + 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.424621 - 0.158335i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.424621 - 0.158335i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (10.3 - 4.28i)T \) |
| 7 | \( 1 + (16.5 - 8.39i)T \) |
| good | 2 | \( 1 + (4.61 + 0.241i)T + (7.95 + 0.836i)T^{2} \) |
| 3 | \( 1 + (-1.85 - 2.29i)T + (-5.61 + 26.4i)T^{2} \) |
| 11 | \( 1 + (-12.7 + 2.71i)T + (1.21e3 - 541. i)T^{2} \) |
| 13 | \( 1 + (54.2 + 27.6i)T + (1.29e3 + 1.77e3i)T^{2} \) |
| 17 | \( 1 + (-27.3 - 71.3i)T + (-3.65e3 + 3.28e3i)T^{2} \) |
| 19 | \( 1 + (-3.44 - 32.7i)T + (-6.70e3 + 1.42e3i)T^{2} \) |
| 23 | \( 1 + (-0.176 + 3.36i)T + (-1.21e4 - 1.27e3i)T^{2} \) |
| 29 | \( 1 + (39.3 - 54.2i)T + (-7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-111. + 249. i)T + (-1.99e4 - 2.21e4i)T^{2} \) |
| 37 | \( 1 + (-200. + 130. i)T + (2.06e4 - 4.62e4i)T^{2} \) |
| 41 | \( 1 + (-354. + 115. i)T + (5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + (251. + 251. i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + (-402. - 154. i)T + (7.71e4 + 6.94e4i)T^{2} \) |
| 53 | \( 1 + (455. - 368. i)T + (3.09e4 - 1.45e5i)T^{2} \) |
| 59 | \( 1 + (-302. + 335. i)T + (-2.14e4 - 2.04e5i)T^{2} \) |
| 61 | \( 1 + (-320. + 288. i)T + (2.37e4 - 2.25e5i)T^{2} \) |
| 67 | \( 1 + (-43.9 + 16.8i)T + (2.23e5 - 2.01e5i)T^{2} \) |
| 71 | \( 1 + (87.7 + 63.7i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-4.65 + 7.16i)T + (-1.58e5 - 3.55e5i)T^{2} \) |
| 79 | \( 1 + (80.1 + 179. i)T + (-3.29e5 + 3.66e5i)T^{2} \) |
| 83 | \( 1 + (-76.1 + 480. i)T + (-5.43e5 - 1.76e5i)T^{2} \) |
| 89 | \( 1 + (103. + 114. i)T + (-7.36e4 + 7.01e5i)T^{2} \) |
| 97 | \( 1 + (-48.3 - 305. i)T + (-8.68e5 + 2.82e5i)T^{2} \) |
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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
−11.94645341654321201538198934721, −10.75215326685813890894056169859, −9.881047324990058401923495424221, −9.256569556098918036290548302696, −8.202621540907179782891352317703, −7.34464313655892987259230590860, −6.20099044474994205665195237330, −3.93536884489887145786368715548, −2.67451455455544107504545507196, −0.44561815509314583255583088748,
0.958477582637067680815211833952, 2.71391087817698486718601806270, 4.62174280098992441348600087143, 6.87681848174228235728126944337, 7.35597727447212039998661547635, 8.255730082440652051897233685704, 9.310353231041596471635318673382, 10.01054790913820241582654772024, 11.20822609134682940403327360747, 12.11082728193748672727585367409
