L(s) = 1 | + (0.238 − 1.39i)2-s + (−1.88 − 0.663i)4-s + (−0.0805 + 0.611i)5-s + (2.18 + 0.584i)7-s + (−1.37 + 2.47i)8-s + (0.833 + 0.257i)10-s + (0.118 − 0.154i)11-s + (0.882 − 0.677i)13-s + (1.33 − 2.90i)14-s + (3.11 + 2.50i)16-s − 2.51i·17-s + (1.37 + 0.568i)19-s + (0.557 − 1.10i)20-s + (−0.186 − 0.201i)22-s + (2.32 − 0.624i)23-s + ⋯ |
L(s) = 1 | + (0.168 − 0.985i)2-s + (−0.943 − 0.331i)4-s + (−0.0360 + 0.273i)5-s + (0.824 + 0.220i)7-s + (−0.485 + 0.874i)8-s + (0.263 + 0.0815i)10-s + (0.0356 − 0.0465i)11-s + (0.244 − 0.187i)13-s + (0.356 − 0.775i)14-s + (0.779 + 0.626i)16-s − 0.609i·17-s + (0.315 + 0.130i)19-s + (0.124 − 0.246i)20-s + (−0.0398 − 0.0430i)22-s + (0.485 − 0.130i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.35972 - 1.02076i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.35972 - 1.02076i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.238 + 1.39i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.0805 - 0.611i)T + (-4.82 - 1.29i)T^{2} \) |
| 7 | \( 1 + (-2.18 - 0.584i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-0.118 + 0.154i)T + (-2.84 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-0.882 + 0.677i)T + (3.36 - 12.5i)T^{2} \) |
| 17 | \( 1 + 2.51iT - 17T^{2} \) |
| 19 | \( 1 + (-1.37 - 0.568i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-2.32 + 0.624i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-9.87 + 1.29i)T + (28.0 - 7.50i)T^{2} \) |
| 31 | \( 1 + (-1.77 + 3.07i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (6.25 - 2.59i)T + (26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (-0.164 + 0.0442i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-3.05 + 3.98i)T + (-11.1 - 41.5i)T^{2} \) |
| 47 | \( 1 + (-6.51 + 3.76i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.44 + 5.90i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (0.957 - 7.27i)T + (-56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (11.8 - 1.56i)T + (58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (-2.90 - 3.78i)T + (-17.3 + 64.7i)T^{2} \) |
| 71 | \( 1 + (2.34 - 2.34i)T - 71iT^{2} \) |
| 73 | \( 1 + (8.46 + 8.46i)T + 73iT^{2} \) |
| 79 | \( 1 + (-8.87 + 5.12i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.750 - 5.70i)T + (-80.1 + 21.4i)T^{2} \) |
| 89 | \( 1 + (-9.23 + 9.23i)T - 89iT^{2} \) |
| 97 | \( 1 + (-8.80 - 15.2i)T + (-48.5 + 84.0i)T^{2} \) |
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
−10.33598065979673410422147553899, −9.170205358516131603390680500429, −8.558920816619535767497428009228, −7.62244119748561314030799673338, −6.40638895794039983412593601198, −5.23965899584021918826236372551, −4.60758920324134761862024474983, −3.36015656687471341617055439233, −2.41628864005323443350791859929, −1.05026878707371286488534296766,
1.19106397292009139329758816180, 3.13559951377210802658252474881, 4.43985677426228884281248573658, 4.94259454689620682767815475806, 6.06005039100887154101453345828, 6.90487040105133359249720527692, 7.79467553314860879890878043739, 8.535389390382889912370406763552, 9.141644568718249466938689650582, 10.28690611902286381367830120691