| L(s) = 1 | + (1.09 − 0.894i)2-s + (1.24 + 1.33i)3-s + (0.399 − 1.95i)4-s + (−1.36 − 3.00i)5-s + (2.55 + 0.342i)6-s + (−0.257 + 2.63i)7-s + (−1.31 − 2.50i)8-s + (−0.0212 + 0.323i)9-s + (−4.17 − 2.07i)10-s + (−1.13 − 1.82i)11-s + (3.10 − 1.91i)12-s + (−0.322 − 3.27i)13-s + (2.07 + 3.11i)14-s + (2.30 − 5.55i)15-s + (−3.68 − 1.56i)16-s + (2.68 + 0.353i)17-s + ⋯ |
| L(s) = 1 | + (0.774 − 0.632i)2-s + (0.719 + 0.768i)3-s + (0.199 − 0.979i)4-s + (−0.608 − 1.34i)5-s + (1.04 + 0.139i)6-s + (−0.0971 + 0.995i)7-s + (−0.464 − 0.885i)8-s + (−0.00707 + 0.107i)9-s + (−1.32 − 0.655i)10-s + (−0.343 − 0.551i)11-s + (0.896 − 0.551i)12-s + (−0.0895 − 0.908i)13-s + (0.554 + 0.832i)14-s + (0.594 − 1.43i)15-s + (−0.920 − 0.391i)16-s + (0.651 + 0.0858i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.230 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.56081 - 1.97457i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.56081 - 1.97457i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.09 + 0.894i)T \) |
| 7 | \( 1 + (0.257 - 2.63i)T \) |
| good | 3 | \( 1 + (-1.24 - 1.33i)T + (-0.196 + 2.99i)T^{2} \) |
| 5 | \( 1 + (1.36 + 3.00i)T + (-3.29 + 3.75i)T^{2} \) |
| 11 | \( 1 + (1.13 + 1.82i)T + (-4.86 + 9.86i)T^{2} \) |
| 13 | \( 1 + (0.322 + 3.27i)T + (-12.7 + 2.53i)T^{2} \) |
| 17 | \( 1 + (-2.68 - 0.353i)T + (16.4 + 4.39i)T^{2} \) |
| 19 | \( 1 + (-0.843 + 5.10i)T + (-17.9 - 6.10i)T^{2} \) |
| 23 | \( 1 + (-5.30 - 6.05i)T + (-3.00 + 22.8i)T^{2} \) |
| 29 | \( 1 + (3.59 + 6.72i)T + (-16.1 + 24.1i)T^{2} \) |
| 31 | \( 1 + (4.03 - 1.08i)T + (26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (-0.394 + 1.04i)T + (-27.8 - 24.3i)T^{2} \) |
| 41 | \( 1 + (-1.50 - 7.56i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-9.37 + 2.84i)T + (35.7 - 23.8i)T^{2} \) |
| 47 | \( 1 + (0.676 - 0.882i)T + (-12.1 - 45.3i)T^{2} \) |
| 53 | \( 1 + (-2.82 - 4.53i)T + (-23.4 + 47.5i)T^{2} \) |
| 59 | \( 1 + (5.36 - 7.48i)T + (-18.9 - 55.8i)T^{2} \) |
| 61 | \( 1 + (8.98 - 2.09i)T + (54.7 - 26.9i)T^{2} \) |
| 67 | \( 1 + (-9.88 - 10.5i)T + (-4.38 + 66.8i)T^{2} \) |
| 71 | \( 1 + (-7.27 - 10.8i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-6.37 + 3.14i)T + (44.4 - 57.9i)T^{2} \) |
| 79 | \( 1 + (1.05 + 8.00i)T + (-76.3 + 20.4i)T^{2} \) |
| 83 | \( 1 + (-4.91 + 5.98i)T + (-16.1 - 81.4i)T^{2} \) |
| 89 | \( 1 + (-1.18 - 3.50i)T + (-70.6 + 54.1i)T^{2} \) |
| 97 | \( 1 + (-2.57 + 2.57i)T - 97iT^{2} \) |
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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
−9.611439627768021620545627656764, −9.247883450840988211583266447568, −8.523848593723996456043290288004, −7.52942307981272296266842678941, −5.82780454927601349630519053166, −5.27414273761708793992432344393, −4.41138307231238224691253699317, −3.42320219969221733618494524125, −2.68446432700923211254059221755, −0.879956088922235044628090776106,
2.06996939822646254731553023365, 3.17754145204675223277972577481, 3.85298863736665446804744248974, 5.01325664807236872959714476226, 6.47466624435477996298382429063, 7.08269436661328914982217344506, 7.51035777496102830430315503091, 8.129653144509771814609811091543, 9.359383815032437140222238343589, 10.72535109618437901017837165894
