| L(s) = 1 | + (−1.59 − 0.0837i)2-s + (−4.11 − 5.08i)3-s + (−5.40 − 0.568i)4-s + (−9.89 + 5.20i)5-s + (6.15 + 8.46i)6-s + (−14.4 + 11.5i)7-s + (21.2 + 3.36i)8-s + (−3.27 + 15.3i)9-s + (16.2 − 7.49i)10-s + (−56.8 + 12.0i)11-s + (19.3 + 29.8i)12-s + (−10.5 − 5.36i)13-s + (24.0 − 17.2i)14-s + (67.1 + 28.8i)15-s + (8.85 + 1.88i)16-s + (−7.45 − 19.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.565 − 0.0296i)2-s + (−0.791 − 0.977i)3-s + (−0.675 − 0.0710i)4-s + (−0.884 + 0.465i)5-s + (0.418 + 0.576i)6-s + (−0.781 + 0.624i)7-s + (0.939 + 0.148i)8-s + (−0.121 + 0.570i)9-s + (0.514 − 0.236i)10-s + (−1.55 + 0.331i)11-s + (0.465 + 0.717i)12-s + (−0.224 − 0.114i)13-s + (0.460 − 0.329i)14-s + (1.15 + 0.496i)15-s + (0.138 + 0.0294i)16-s + (−0.106 − 0.276i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.255809 - 0.0421656i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.255809 - 0.0421656i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (9.89 - 5.20i)T \) |
| 7 | \( 1 + (14.4 - 11.5i)T \) |
| good | 2 | \( 1 + (1.59 + 0.0837i)T + (7.95 + 0.836i)T^{2} \) |
| 3 | \( 1 + (4.11 + 5.08i)T + (-5.61 + 26.4i)T^{2} \) |
| 11 | \( 1 + (56.8 - 12.0i)T + (1.21e3 - 541. i)T^{2} \) |
| 13 | \( 1 + (10.5 + 5.36i)T + (1.29e3 + 1.77e3i)T^{2} \) |
| 17 | \( 1 + (7.45 + 19.4i)T + (-3.65e3 + 3.28e3i)T^{2} \) |
| 19 | \( 1 + (8.62 + 82.0i)T + (-6.70e3 + 1.42e3i)T^{2} \) |
| 23 | \( 1 + (0.885 - 16.8i)T + (-1.21e4 - 1.27e3i)T^{2} \) |
| 29 | \( 1 + (-93.2 + 128. i)T + (-7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-2.66 + 5.98i)T + (-1.99e4 - 2.21e4i)T^{2} \) |
| 37 | \( 1 + (210. - 136. i)T + (2.06e4 - 4.62e4i)T^{2} \) |
| 41 | \( 1 + (-262. + 85.2i)T + (5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + (-313. - 313. i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + (-305. - 117. i)T + (7.71e4 + 6.94e4i)T^{2} \) |
| 53 | \( 1 + (371. - 301. i)T + (3.09e4 - 1.45e5i)T^{2} \) |
| 59 | \( 1 + (376. - 418. i)T + (-2.14e4 - 2.04e5i)T^{2} \) |
| 61 | \( 1 + (-32.2 + 29.0i)T + (2.37e4 - 2.25e5i)T^{2} \) |
| 67 | \( 1 + (160. - 61.6i)T + (2.23e5 - 2.01e5i)T^{2} \) |
| 71 | \( 1 + (212. + 154. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-332. + 511. i)T + (-1.58e5 - 3.55e5i)T^{2} \) |
| 79 | \( 1 + (105. + 237. i)T + (-3.29e5 + 3.66e5i)T^{2} \) |
| 83 | \( 1 + (200. - 1.26e3i)T + (-5.43e5 - 1.76e5i)T^{2} \) |
| 89 | \( 1 + (621. + 690. i)T + (-7.36e4 + 7.01e5i)T^{2} \) |
| 97 | \( 1 + (160. + 1.01e3i)T + (-8.68e5 + 2.82e5i)T^{2} \) |
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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
−12.34123963695066235573412818410, −11.19990432308991643304970278270, −10.29071944958069824844962003962, −9.128526588754261880779683482729, −7.87776821303969113251561776373, −7.20895560871345278265534045760, −5.93520802773859395724745251533, −4.66569112503282508364772755407, −2.72057945850257021327463892021, −0.48069041928634802393870108664,
0.39413555068805695081450345532, 3.64082799771803796745804437039, 4.59074842869579561965501993061, 5.55826495567651534317814965535, 7.39401616703320198923690918842, 8.297767837102523137437634650557, 9.433310023492391645976565993054, 10.48510764659891912629988288712, 10.75255844073591600161194335430, 12.34174495907763076852519744354
