| L(s) = 1 | + (−0.373 + 1.75i)2-s + (−2.87 − 0.301i)3-s + (−1.11 − 0.496i)4-s + (−1.49 + 1.34i)5-s + (1.60 − 4.93i)6-s + (−1.89 − 1.84i)7-s + (−0.822 + 1.13i)8-s + (5.22 + 1.11i)9-s + (−1.80 − 3.12i)10-s + (−0.00719 + 3.31i)11-s + (3.05 + 1.76i)12-s + (1.45 + 4.46i)13-s + (3.95 − 2.63i)14-s + (4.70 − 3.41i)15-s + (−3.31 − 3.67i)16-s + (−0.563 + 0.119i)17-s + ⋯ |
| L(s) = 1 | + (−0.263 + 1.24i)2-s + (−1.65 − 0.174i)3-s + (−0.557 − 0.248i)4-s + (−0.668 + 0.602i)5-s + (0.654 − 2.01i)6-s + (−0.715 − 0.698i)7-s + (−0.290 + 0.400i)8-s + (1.74 + 0.370i)9-s + (−0.570 − 0.988i)10-s + (−0.00216 + 0.999i)11-s + (0.881 + 0.508i)12-s + (0.402 + 1.23i)13-s + (1.05 − 0.703i)14-s + (1.21 − 0.882i)15-s + (−0.828 − 0.919i)16-s + (−0.136 + 0.0290i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.165i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0265155 - 0.317768i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0265155 - 0.317768i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (1.89 + 1.84i)T \) |
| 11 | \( 1 + (0.00719 - 3.31i)T \) |
| good | 2 | \( 1 + (0.373 - 1.75i)T + (-1.82 - 0.813i)T^{2} \) |
| 3 | \( 1 + (2.87 + 0.301i)T + (2.93 + 0.623i)T^{2} \) |
| 5 | \( 1 + (1.49 - 1.34i)T + (0.522 - 4.97i)T^{2} \) |
| 13 | \( 1 + (-1.45 - 4.46i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (0.563 - 0.119i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (-1.25 + 0.560i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-0.408 + 0.707i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.425 - 0.585i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (2.66 + 2.40i)T + (3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (1.01 + 9.64i)T + (-36.1 + 7.69i)T^{2} \) |
| 41 | \( 1 + (-2.23 - 1.62i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 8.37iT - 43T^{2} \) |
| 47 | \( 1 + (-1.44 - 3.23i)T + (-31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (4.28 - 4.75i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (2.32 - 5.21i)T + (-39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-9.07 - 10.0i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (0.838 + 1.45i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.10 + 6.46i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-5.68 - 2.53i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (1.58 - 7.47i)T + (-72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (0.369 - 1.13i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (12.6 + 7.31i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.81 + 3.18i)T + (78.4 - 57.0i)T^{2} \) |
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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
−15.50313209046784169377608904592, −14.29062046041719579822127795411, −12.76874838525294344667474128298, −11.63251422280599670266831841628, −10.88028084812813065826174297673, −9.449619302030225010388411302134, −7.38705058820390402480259189771, −6.93703771474712675882312677900, −5.97222608873359029193492077499, −4.38976228136895281771499247109,
0.54582119518684852596397437725, 3.48870129545204739543173607402, 5.31202100150229258654618889868, 6.41237354439191435429398534383, 8.534437419543048130168662030703, 9.938844766646226052994025602117, 10.85713329757776906443553471658, 11.69117310946294522358191366228, 12.35786366801943053528323693554, 13.12580528946631789666196466791
