| L(s) = 1 | + (−3.23 + 2.35i)2-s + (2.51 − 23.9i)3-s + (4.94 − 15.2i)4-s + (54.0 − 93.5i)5-s + (48.1 + 83.3i)6-s + (77.2 − 16.4i)7-s + (19.7 + 60.8i)8-s + (−328. − 69.7i)9-s + (45.1 + 429. i)10-s + (38.9 + 43.2i)11-s + (−351. − 156. i)12-s + (281. − 125. i)13-s + (−211. + 234. i)14-s + (−2.10e3 − 1.52e3i)15-s + (−207. − 150. i)16-s + (−461. + 512. i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (0.161 − 1.53i)3-s + (0.154 − 0.475i)4-s + (0.966 − 1.67i)5-s + (0.545 + 0.945i)6-s + (0.595 − 0.126i)7-s + (0.109 + 0.336i)8-s + (−1.35 − 0.287i)9-s + (0.142 + 1.35i)10-s + (0.0971 + 0.107i)11-s + (−0.704 − 0.313i)12-s + (0.461 − 0.205i)13-s + (−0.288 + 0.319i)14-s + (−2.41 − 1.75i)15-s + (−0.202 − 0.146i)16-s + (−0.387 + 0.430i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.573 + 0.818i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.573 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.765172 - 1.47077i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.765172 - 1.47077i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.23 - 2.35i)T \) |
| 31 | \( 1 + (4.58e3 - 2.75e3i)T \) |
| good | 3 | \( 1 + (-2.51 + 23.9i)T + (-237. - 50.5i)T^{2} \) |
| 5 | \( 1 + (-54.0 + 93.5i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-77.2 + 16.4i)T + (1.53e4 - 6.83e3i)T^{2} \) |
| 11 | \( 1 + (-38.9 - 43.2i)T + (-1.68e4 + 1.60e5i)T^{2} \) |
| 13 | \( 1 + (-281. + 125. i)T + (2.48e5 - 2.75e5i)T^{2} \) |
| 17 | \( 1 + (461. - 512. i)T + (-1.48e5 - 1.41e6i)T^{2} \) |
| 19 | \( 1 + (-2.08e3 - 930. i)T + (1.65e6 + 1.84e6i)T^{2} \) |
| 23 | \( 1 + (16.2 + 50.1i)T + (-5.20e6 + 3.78e6i)T^{2} \) |
| 29 | \( 1 + (2.36e3 - 1.72e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 37 | \( 1 + (-6.78e3 - 1.17e4i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + (-1.62e3 - 1.54e4i)T + (-1.13e8 + 2.40e7i)T^{2} \) |
| 43 | \( 1 + (-8.75e3 - 3.89e3i)T + (9.83e7 + 1.09e8i)T^{2} \) |
| 47 | \( 1 + (2.17e4 + 1.57e4i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-3.56e4 - 7.58e3i)T + (3.82e8 + 1.70e8i)T^{2} \) |
| 59 | \( 1 + (-4.54e3 + 4.32e4i)T + (-6.99e8 - 1.48e8i)T^{2} \) |
| 61 | \( 1 - 181.T + 8.44e8T^{2} \) |
| 67 | \( 1 + (-5.78e3 + 1.00e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-2.98e4 - 6.34e3i)T + (1.64e9 + 7.33e8i)T^{2} \) |
| 73 | \( 1 + (-3.62e4 - 4.02e4i)T + (-2.16e8 + 2.06e9i)T^{2} \) |
| 79 | \( 1 + (3.42e3 - 3.80e3i)T + (-3.21e8 - 3.06e9i)T^{2} \) |
| 83 | \( 1 + (1.15e4 + 1.10e5i)T + (-3.85e9 + 8.18e8i)T^{2} \) |
| 89 | \( 1 + (-1.98e4 + 6.09e4i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (3.33e4 - 1.02e5i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−13.43078042481029823520406170501, −12.79205173180586203592413794666, −11.61252241514980662885629204331, −9.723609298647071750549117684547, −8.561994858972878509013113647102, −7.83621545866777467138380317839, −6.31458724852000482383280770357, −5.17381248879402762878373335155, −1.72732866115561572114706890157, −1.02602316919556917356503072408,
2.41986527254880065294742135797, 3.70471786322561789271014069053, 5.59527096457639161778697807866, 7.27736404109562988528592045092, 9.152539872036082218078045158984, 9.799830545402515142850541381505, 10.93286621749229240218458266675, 11.27969859238678931547006887370, 13.65754699867533450594154395066, 14.53721427395362562925761670173
