| 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
−14.53721427395362562925761670173, −13.65754699867533450594154395066, −11.27969859238678931547006887370, −10.93286621749229240218458266675, −9.799830545402515142850541381505, −9.152539872036082218078045158984, −7.27736404109562988528592045092, −5.59527096457639161778697807866, −3.70471786322561789271014069053, −2.41986527254880065294742135797,
1.02602316919556917356503072408, 1.72732866115561572114706890157, 5.17381248879402762878373335155, 6.31458724852000482383280770357, 7.83621545866777467138380317839, 8.561994858972878509013113647102, 9.723609298647071750549117684547, 11.61252241514980662885629204331, 12.79205173180586203592413794666, 13.43078042481029823520406170501
