| L(s) = 1 | + (2.28 − 1.65i)2-s + (7.91 − 24.3i)3-s + (−7.42 + 22.8i)4-s + (−0.762 − 55.8i)5-s + (−22.3 − 68.7i)6-s − 43.3·7-s + (48.8 + 150. i)8-s + (−334. − 242. i)9-s + (−94.5 − 126. i)10-s + (545. − 396. i)11-s + (497. + 361. i)12-s + (261. + 189. i)13-s + (−99.0 + 71.9i)14-s + (−1.36e3 − 423. i)15-s + (−260. − 189. i)16-s + (648. + 1.99e3i)17-s + ⋯ |
| L(s) = 1 | + (0.403 − 0.293i)2-s + (0.507 − 1.56i)3-s + (−0.232 + 0.714i)4-s + (−0.0136 − 0.999i)5-s + (−0.253 − 0.779i)6-s − 0.334·7-s + (0.270 + 0.831i)8-s + (−1.37 − 0.998i)9-s + (−0.298 − 0.399i)10-s + (1.35 − 0.987i)11-s + (0.997 + 0.725i)12-s + (0.428 + 0.311i)13-s + (−0.135 + 0.0981i)14-s + (−1.56 − 0.486i)15-s + (−0.254 − 0.184i)16-s + (0.544 + 1.67i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.160 + 0.987i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.24929 - 1.46886i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.24929 - 1.46886i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.762 + 55.8i)T \) |
| good | 2 | \( 1 + (-2.28 + 1.65i)T + (9.88 - 30.4i)T^{2} \) |
| 3 | \( 1 + (-7.91 + 24.3i)T + (-196. - 142. i)T^{2} \) |
| 7 | \( 1 + 43.3T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-545. + 396. i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-261. - 189. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-648. - 1.99e3i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-317. - 976. i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-1.20e3 + 876. i)T + (1.98e6 - 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-654. + 2.01e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-319. - 984. i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-4.66e3 - 3.39e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (1.43e4 + 1.04e4i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 + 4.14e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (4.08e3 - 1.25e4i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (4.88e3 - 1.50e4i)T + (-3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-7.54e3 - 5.48e3i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-2.25e3 + 1.64e3i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (218. + 673. i)T + (-1.09e9 + 7.93e8i)T^{2} \) |
| 71 | \( 1 + (2.40e4 - 7.41e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-5.59e4 + 4.06e4i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-1.17e4 + 3.62e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-7.89e3 - 2.43e4i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + (-6.06e4 + 4.40e4i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-4.80e3 + 1.47e4i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−16.63555479085348777532298511229, −14.29666371884872318688405901770, −13.37828100517438036925718529126, −12.54938259832956130150818857332, −11.73618955805009330521612354235, −8.818965611823172819808589523634, −8.083416914576591391106574954306, −6.25327438564114307861823250811, −3.64949633037211230129792935049, −1.40274337689440476140896120740,
3.44559721735191343697759776917, 4.92265778742368590446764917526, 6.79872609013526368934183295556, 9.391793020062158127027969773399, 9.950475968911032075406519765824, 11.34373024208663027052145151285, 13.76158281396370563704703506781, 14.69416932329631154221750255689, 15.28423255913559805040963724990, 16.35767955212824589127461811428
