| L(s) = 1 | + (−1.54 − 0.992i)2-s + (0.350 + 2.43i)3-s + (0.568 + 1.24i)4-s + (−0.959 − 0.281i)5-s + (1.87 − 4.11i)6-s + (−1.89 + 2.18i)7-s + (−0.165 + 1.14i)8-s + (−2.94 + 0.865i)9-s + (1.20 + 1.38i)10-s + (−2.98 + 1.91i)11-s + (−2.83 + 1.82i)12-s + (2.92 + 3.37i)13-s + (5.08 − 1.49i)14-s + (0.350 − 2.43i)15-s + (3.18 − 3.67i)16-s + (1.20 − 2.64i)17-s + ⋯ |
| L(s) = 1 | + (−1.09 − 0.701i)2-s + (0.202 + 1.40i)3-s + (0.284 + 0.621i)4-s + (−0.429 − 0.125i)5-s + (0.766 − 1.67i)6-s + (−0.715 + 0.825i)7-s + (−0.0584 + 0.406i)8-s + (−0.982 + 0.288i)9-s + (0.380 + 0.438i)10-s + (−0.899 + 0.577i)11-s + (−0.818 + 0.525i)12-s + (0.810 + 0.935i)13-s + (1.35 − 0.399i)14-s + (0.0905 − 0.629i)15-s + (0.796 − 0.919i)16-s + (0.292 − 0.641i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0394 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0394 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.331152 + 0.344488i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.331152 + 0.344488i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.959 + 0.281i)T \) |
| 23 | \( 1 + (4.74 + 0.687i)T \) |
| good | 2 | \( 1 + (1.54 + 0.992i)T + (0.830 + 1.81i)T^{2} \) |
| 3 | \( 1 + (-0.350 - 2.43i)T + (-2.87 + 0.845i)T^{2} \) |
| 7 | \( 1 + (1.89 - 2.18i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (2.98 - 1.91i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.92 - 3.37i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.20 + 2.64i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-1.45 - 3.18i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-3.55 + 7.79i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.0142 + 0.0992i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-3.01 + 0.884i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-10.2 - 3.00i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (0.893 + 6.21i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 1.20T + 47T^{2} \) |
| 53 | \( 1 + (6.99 - 8.06i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-7.51 - 8.67i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (0.656 - 4.56i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-9.45 - 6.07i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (4.50 + 2.89i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (4.77 + 10.4i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-0.504 - 0.582i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (1.75 - 0.514i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-2.50 - 17.4i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (9.15 + 2.68i)T + (81.6 + 52.4i)T^{2} \) |
| 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
−13.98043581209564709897670180855, −12.32450748407994946495437377246, −11.45025932935054747785819959691, −10.34631062952186583018191960276, −9.663568303039423090116371775584, −8.979628820634403102375915668607, −7.889745354386905158473168146686, −5.73756743591902971873991361860, −4.22838127596457971478204870944, −2.67670303846290616963332343460,
0.74402138476839823488162618940, 3.34171748658055360040675120627, 6.06861035414473873537379978960, 7.00967150013005594726530618885, 7.87472328966975233183495713518, 8.433669836738215355864642760332, 9.993123630371841237842240314481, 11.00288557887104978394340434029, 12.74354096201363138015324166644, 13.05242323536605867777309813788
