L(s) = 1 | + (3.36 − 2.16i)2-s + (6.63 − 14.5i)4-s − 43.0·5-s − 14.1i·7-s + (−9.18 − 63.3i)8-s + (−144. + 93.1i)10-s + 209. i·11-s − 195.·13-s + (−30.5 − 47.4i)14-s + (−167. − 193. i)16-s − 303.·17-s − 274. i·19-s + (−285. + 626. i)20-s + (453. + 704. i)22-s − 466. i·23-s + ⋯ |
L(s) = 1 | + (0.841 − 0.540i)2-s + (0.414 − 0.909i)4-s − 1.72·5-s − 0.288i·7-s + (−0.143 − 0.989i)8-s + (−1.44 + 0.931i)10-s + 1.73i·11-s − 1.15·13-s + (−0.155 − 0.242i)14-s + (−0.656 − 0.754i)16-s − 1.05·17-s − 0.761i·19-s + (−0.714 + 1.56i)20-s + (0.936 + 1.45i)22-s − 0.881i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 - 0.414i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.909 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.104996 + 0.483583i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.104996 + 0.483583i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-3.36 + 2.16i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 43.0T + 625T^{2} \) |
| 7 | \( 1 + 14.1iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 209. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 195.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 303.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 274. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 466. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 439.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.22e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 624.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 203.T + 2.82e6T^{2} \) |
| 43 | \( 1 + 419. iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 1.99e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 206.T + 7.89e6T^{2} \) |
| 59 | \( 1 + 2.32e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 1.39e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 7.34e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 8.54e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 3.24e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 6.15e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 213. iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 8.43e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 1.02e3T + 8.85e7T^{2} \) |
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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
−12.32336601648193445219447684534, −11.60267763859544395137744671758, −10.61219473704800953828945429819, −9.371194123067764937671281934791, −7.58796696938285327833992958565, −6.87634026198035650117560459666, −4.69250779000836874929325407264, −4.20416590340475788526054293121, −2.46363680760086784172211193320, −0.15863656552820087026755668749,
3.07648674690865398081628016939, 4.12062893023030389456105580022, 5.46620297252736245817391043635, 6.93849255350193216306503593680, 7.947230808525174524942889841820, 8.761015127851082360788251854752, 10.96498452387400413376552702938, 11.69176705422377020446731570164, 12.46030950601739772502375178400, 13.66485321247732278917401542076