| L(s) = 1 | + (0.998 + 0.0475i)2-s + (−1.50 + 0.864i)3-s + (0.995 + 0.0950i)4-s + (1.85 − 1.77i)5-s + (−1.54 + 0.791i)6-s + (−0.441 + 2.28i)7-s + (0.989 + 0.142i)8-s + (1.50 − 2.59i)9-s + (1.94 − 1.68i)10-s + (3.18 + 1.64i)11-s + (−1.57 + 0.717i)12-s + (−1.34 + 0.259i)13-s + (−0.549 + 2.26i)14-s + (−1.25 + 4.26i)15-s + (0.981 + 0.189i)16-s + (0.749 − 1.64i)17-s + ⋯ |
| L(s) = 1 | + (0.706 + 0.0336i)2-s + (−0.866 + 0.498i)3-s + (0.497 + 0.0475i)4-s + (0.831 − 0.793i)5-s + (−0.628 + 0.323i)6-s + (−0.166 + 0.865i)7-s + (0.349 + 0.0503i)8-s + (0.502 − 0.864i)9-s + (0.614 − 0.532i)10-s + (0.960 + 0.494i)11-s + (−0.455 + 0.207i)12-s + (−0.372 + 0.0718i)13-s + (−0.146 + 0.605i)14-s + (−0.325 + 1.10i)15-s + (0.245 + 0.0473i)16-s + (0.181 − 0.398i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.84590 + 0.402781i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.84590 + 0.402781i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.998 - 0.0475i)T \) |
| 3 | \( 1 + (1.50 - 0.864i)T \) |
| 23 | \( 1 + (-4.69 - 0.954i)T \) |
| good | 5 | \( 1 + (-1.85 + 1.77i)T + (0.237 - 4.99i)T^{2} \) |
| 7 | \( 1 + (0.441 - 2.28i)T + (-6.49 - 2.60i)T^{2} \) |
| 11 | \( 1 + (-3.18 - 1.64i)T + (6.38 + 8.96i)T^{2} \) |
| 13 | \( 1 + (1.34 - 0.259i)T + (12.0 - 4.83i)T^{2} \) |
| 17 | \( 1 + (-0.749 + 1.64i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-2.19 + 1.00i)T + (12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-0.682 - 7.15i)T + (-28.4 + 5.48i)T^{2} \) |
| 31 | \( 1 + (-1.71 - 1.35i)T + (7.30 + 30.1i)T^{2} \) |
| 37 | \( 1 + (1.75 + 5.96i)T + (-31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (7.43 + 7.79i)T + (-1.95 + 40.9i)T^{2} \) |
| 43 | \( 1 + (3.73 + 4.74i)T + (-10.1 + 41.7i)T^{2} \) |
| 47 | \( 1 + (10.6 - 6.15i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.99 - 2.30i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (1.09 + 5.67i)T + (-54.7 + 21.9i)T^{2} \) |
| 61 | \( 1 + (0.866 + 2.16i)T + (-44.1 + 42.0i)T^{2} \) |
| 67 | \( 1 + (5.94 + 11.5i)T + (-38.8 + 54.5i)T^{2} \) |
| 71 | \( 1 + (-1.35 + 2.10i)T + (-29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (4.61 + 10.1i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (2.17 + 0.752i)T + (62.0 + 48.8i)T^{2} \) |
| 83 | \( 1 + (-1.61 - 1.53i)T + (3.94 + 82.9i)T^{2} \) |
| 89 | \( 1 + (-0.0631 - 0.439i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (3.14 - 0.764i)T + (86.2 - 44.4i)T^{2} \) |
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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
−11.53684318364689245168353851328, −10.45229474312337826942674382480, −9.356405363410264233055022414392, −9.031320849138659768026530503599, −7.13762486375108552232909982118, −6.27899046970098224834674666057, −5.22366908618272703470901747922, −4.90368067306588832293955106055, −3.37087586064605718890226975681, −1.60332667341350154900745620556,
1.40008771715740956627034115390, 2.98937149690414930964189836494, 4.35743530152217059836940287436, 5.55395627727207988633276995206, 6.52738948775581329118427578331, 6.84883591034167694933568699358, 8.094478391066354641596427241448, 9.889401681256131342405709991064, 10.30792666283435823720570517098, 11.40179871187892041703481324972
