| L(s) = 1 | + (−1.24 − 0.675i)2-s + (−1.48 + 0.166i)3-s + (1.08 + 1.67i)4-s + (0.727 − 1.51i)5-s + (1.95 + 0.793i)6-s + (−3.42 − 2.73i)7-s + (−0.214 − 2.82i)8-s + (−0.758 + 0.173i)9-s + (−1.92 + 1.38i)10-s + (1.20 − 0.755i)11-s + (−1.88 − 2.30i)12-s + (−5.83 − 1.33i)13-s + (2.41 + 5.71i)14-s + (−0.825 + 2.35i)15-s + (−1.63 + 3.64i)16-s + (3.39 − 3.39i)17-s + ⋯ |
| L(s) = 1 | + (−0.878 − 0.477i)2-s + (−0.855 + 0.0963i)3-s + (0.543 + 0.839i)4-s + (0.325 − 0.675i)5-s + (0.797 + 0.324i)6-s + (−1.29 − 1.03i)7-s + (−0.0759 − 0.997i)8-s + (−0.252 + 0.0577i)9-s + (−0.608 + 0.437i)10-s + (0.362 − 0.227i)11-s + (−0.545 − 0.665i)12-s + (−1.61 − 0.369i)13-s + (0.644 + 1.52i)14-s + (−0.213 + 0.609i)15-s + (−0.409 + 0.912i)16-s + (0.823 − 0.823i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.818 + 0.574i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.818 + 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.100817 - 0.319436i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.100817 - 0.319436i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.24 + 0.675i)T \) |
| 29 | \( 1 + (2.66 + 4.68i)T \) |
| good | 3 | \( 1 + (1.48 - 0.166i)T + (2.92 - 0.667i)T^{2} \) |
| 5 | \( 1 + (-0.727 + 1.51i)T + (-3.11 - 3.90i)T^{2} \) |
| 7 | \( 1 + (3.42 + 2.73i)T + (1.55 + 6.82i)T^{2} \) |
| 11 | \( 1 + (-1.20 + 0.755i)T + (4.77 - 9.91i)T^{2} \) |
| 13 | \( 1 + (5.83 + 1.33i)T + (11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (-3.39 + 3.39i)T - 17iT^{2} \) |
| 19 | \( 1 + (-0.339 + 3.01i)T + (-18.5 - 4.22i)T^{2} \) |
| 23 | \( 1 + (-2.08 - 4.33i)T + (-14.3 + 17.9i)T^{2} \) |
| 31 | \( 1 + (-0.727 - 2.08i)T + (-24.2 + 19.3i)T^{2} \) |
| 37 | \( 1 + (1.98 - 3.16i)T + (-16.0 - 33.3i)T^{2} \) |
| 41 | \( 1 + (0.423 + 0.423i)T + 41iT^{2} \) |
| 43 | \( 1 + (-0.259 - 0.0908i)T + (33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (-0.934 - 1.48i)T + (-20.3 + 42.3i)T^{2} \) |
| 53 | \( 1 + (0.888 + 0.428i)T + (33.0 + 41.4i)T^{2} \) |
| 59 | \( 1 + 6.65iT - 59T^{2} \) |
| 61 | \( 1 + (1.19 + 10.5i)T + (-59.4 + 13.5i)T^{2} \) |
| 67 | \( 1 + (2.46 + 10.8i)T + (-60.3 + 29.0i)T^{2} \) |
| 71 | \( 1 + (-2.83 + 12.4i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-11.0 - 3.87i)T + (57.0 + 45.5i)T^{2} \) |
| 79 | \( 1 + (4.00 - 6.37i)T + (-34.2 - 71.1i)T^{2} \) |
| 83 | \( 1 + (3.24 - 2.58i)T + (18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-11.5 + 4.03i)T + (69.5 - 55.4i)T^{2} \) |
| 97 | \( 1 + (-0.301 + 2.67i)T + (-94.5 - 21.5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.79127293250174025535226950828, −12.00626487933126751906950959759, −10.95319028400440809574887123790, −9.832586197476621613140259056260, −9.359509260224649012236334476196, −7.62878752022747709295357980480, −6.62342550994617772138228617418, −5.07503384046946821921414690525, −3.16771230381790824553199480265, −0.50893969431236276015554305885,
2.63063758019883934074034025992, 5.46934445041258169304322983739, 6.28819651110332609146032257403, 7.11228586357843697550739645934, 8.792617397976040998483266322224, 9.809754580268196374768912417758, 10.53570943227049492635544828615, 11.93504998759328032378386047775, 12.50055129239169120308394133913, 14.51790675466720558270028308380
