| L(s) = 1 | + (−2.85 + 3.86i)2-s + (−6.75 + 1.27i)3-s + (−4.45 − 14.4i)4-s + (6.74 − 8.91i)5-s + (14.3 − 29.7i)6-s + (−11.2 − 14.7i)7-s + (32.2 + 11.3i)8-s + (18.7 − 7.37i)9-s + (15.2 + 51.5i)10-s + (19.0 − 48.5i)11-s + (48.5 + 91.8i)12-s + (9.34 + 1.05i)13-s + (89.0 − 1.35i)14-s + (−34.1 + 68.7i)15-s + (−36.0 + 24.5i)16-s + (−112. − 4.22i)17-s + ⋯ |
| L(s) = 1 | + (−1.00 + 1.36i)2-s + (−1.29 + 0.245i)3-s + (−0.556 − 1.80i)4-s + (0.603 − 0.797i)5-s + (0.975 − 2.02i)6-s + (−0.605 − 0.795i)7-s + (1.42 + 0.499i)8-s + (0.696 − 0.273i)9-s + (0.481 + 1.63i)10-s + (0.522 − 1.33i)11-s + (1.16 + 2.20i)12-s + (0.199 + 0.0224i)13-s + (1.69 − 0.0258i)14-s + (−0.587 + 1.18i)15-s + (−0.562 + 0.383i)16-s + (−1.61 − 0.0603i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.369i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.00140719 - 0.00734580i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00140719 - 0.00734580i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-6.74 + 8.91i)T \) |
| 7 | \( 1 + (11.2 + 14.7i)T \) |
| good | 2 | \( 1 + (2.85 - 3.86i)T + (-2.35 - 7.64i)T^{2} \) |
| 3 | \( 1 + (6.75 - 1.27i)T + (25.1 - 9.86i)T^{2} \) |
| 11 | \( 1 + (-19.0 + 48.5i)T + (-975. - 905. i)T^{2} \) |
| 13 | \( 1 + (-9.34 - 1.05i)T + (2.14e3 + 488. i)T^{2} \) |
| 17 | \( 1 + (112. + 4.22i)T + (4.89e3 + 367. i)T^{2} \) |
| 19 | \( 1 + (34.5 - 59.8i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-3.35 - 89.7i)T + (-1.21e4 + 909. i)T^{2} \) |
| 29 | \( 1 + (38.1 - 8.70i)T + (2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (-226. + 130. i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (47.2 - 24.9i)T + (2.85e4 - 4.18e4i)T^{2} \) |
| 41 | \( 1 + (178. + 371. i)T + (-4.29e4 + 5.38e4i)T^{2} \) |
| 43 | \( 1 + (72.7 + 208. i)T + (-6.21e4 + 4.95e4i)T^{2} \) |
| 47 | \( 1 + (-280. - 206. i)T + (3.06e4 + 9.92e4i)T^{2} \) |
| 53 | \( 1 + (194. + 102. i)T + (8.38e4 + 1.23e5i)T^{2} \) |
| 59 | \( 1 + (-53.3 - 711. i)T + (-2.03e5 + 3.06e4i)T^{2} \) |
| 61 | \( 1 + (116. - 377. i)T + (-1.87e5 - 1.27e5i)T^{2} \) |
| 67 | \( 1 + (-105. - 28.2i)T + (2.60e5 + 1.50e5i)T^{2} \) |
| 71 | \( 1 + (-34.2 + 150. i)T + (-3.22e5 - 1.55e5i)T^{2} \) |
| 73 | \( 1 + (561. - 414. i)T + (1.14e5 - 3.71e5i)T^{2} \) |
| 79 | \( 1 + (974. + 562. i)T + (2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-87.6 - 778. i)T + (-5.57e5 + 1.27e5i)T^{2} \) |
| 89 | \( 1 + (-225. - 574. i)T + (-5.16e5 + 4.79e5i)T^{2} \) |
| 97 | \( 1 + (-717. - 717. i)T + 9.12e5iT^{2} \) |
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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
−11.98882693062254747220758306961, −10.86047185707064530797740733927, −10.12767303428856821727710112627, −9.112653466983640184833087397845, −8.395515525251760119935337928779, −6.94064216690549389687486419190, −6.13145772148579908297957909111, −5.63163582066715061504921372509, −4.27466026714436377580632704362, −0.975226154691022759642359021523,
0.00634504044986155289354194150, 1.79540375727081202773609704683, 2.82197591871621968978944693811, 4.65779571517865747342351414962, 6.32600369600012273000312619949, 6.83016671043573746442343376235, 8.648954332643265858310036194565, 9.541410508320922749885915552771, 10.30784472257033291879794708044, 11.10966588449276482785176299214
