| L(s) = 1 | + (0.337 − 0.0650i)2-s + (1.00 − 2.20i)3-s + (−1.74 + 0.699i)4-s + (0.701 + 0.205i)5-s + (0.196 − 0.808i)6-s + (−0.183 + 0.531i)7-s + (−1.12 + 0.721i)8-s + (−1.86 − 2.15i)9-s + (0.250 + 0.0238i)10-s + (1.03 + 4.27i)11-s + (−0.216 + 4.54i)12-s + (0.265 − 0.136i)13-s + (−0.0275 + 0.191i)14-s + (1.15 − 1.33i)15-s + (2.39 − 2.28i)16-s + (−6.52 − 2.61i)17-s + ⋯ |
| L(s) = 1 | + (0.238 − 0.0460i)2-s + (0.580 − 1.27i)3-s + (−0.873 + 0.349i)4-s + (0.313 + 0.0921i)5-s + (0.0800 − 0.329i)6-s + (−0.0694 + 0.200i)7-s + (−0.396 + 0.255i)8-s + (−0.622 − 0.718i)9-s + (0.0791 + 0.00755i)10-s + (0.312 + 1.28i)11-s + (−0.0625 + 1.31i)12-s + (0.0736 − 0.0379i)13-s + (−0.00735 + 0.0511i)14-s + (0.299 − 0.345i)15-s + (0.597 − 0.570i)16-s + (−1.58 − 0.633i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.962741 - 0.342023i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.962741 - 0.342023i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 67 | \( 1 + (7.17 - 3.94i)T \) |
| good | 2 | \( 1 + (-0.337 + 0.0650i)T + (1.85 - 0.743i)T^{2} \) |
| 3 | \( 1 + (-1.00 + 2.20i)T + (-1.96 - 2.26i)T^{2} \) |
| 5 | \( 1 + (-0.701 - 0.205i)T + (4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (0.183 - 0.531i)T + (-5.50 - 4.32i)T^{2} \) |
| 11 | \( 1 + (-1.03 - 4.27i)T + (-9.77 + 5.04i)T^{2} \) |
| 13 | \( 1 + (-0.265 + 0.136i)T + (7.54 - 10.5i)T^{2} \) |
| 17 | \( 1 + (6.52 + 2.61i)T + (12.3 + 11.7i)T^{2} \) |
| 19 | \( 1 + (-1.09 - 3.16i)T + (-14.9 + 11.7i)T^{2} \) |
| 23 | \( 1 + (2.61 + 3.67i)T + (-7.52 + 21.7i)T^{2} \) |
| 29 | \( 1 + (2.28 + 3.94i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-9.09 - 4.68i)T + (17.9 + 25.2i)T^{2} \) |
| 37 | \( 1 + (-2.77 + 4.80i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.82 - 3.01i)T + (9.66 - 39.8i)T^{2} \) |
| 43 | \( 1 + (0.565 + 3.93i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + (0.768 - 0.0733i)T + (46.1 - 8.89i)T^{2} \) |
| 53 | \( 1 + (0.712 - 4.95i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (8.35 - 5.37i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-3.23 + 13.3i)T + (-54.2 - 27.9i)T^{2} \) |
| 71 | \( 1 + (-9.03 + 3.61i)T + (51.3 - 48.9i)T^{2} \) |
| 73 | \( 1 + (2.76 - 11.4i)T + (-64.8 - 33.4i)T^{2} \) |
| 79 | \( 1 + (0.0260 - 0.546i)T + (-78.6 - 7.50i)T^{2} \) |
| 83 | \( 1 + (1.02 - 0.978i)T + (3.94 - 82.9i)T^{2} \) |
| 89 | \( 1 + (-2.97 - 6.50i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-3.45 + 5.98i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−14.19658787610878444508967009382, −13.64757526760472588384014057022, −12.65435335964830832071538798462, −11.96758424270126706886754459666, −9.890602885928405514267176819732, −8.745515784670167905365204154381, −7.66387657649591682342873887836, −6.39669708443376140738076365490, −4.45386781637758068870356733129, −2.34483946133509250904748872239,
3.54901383919862478834764872825, 4.65570046808172512608464966147, 6.09055044232021254532279785154, 8.493120991370942067484982205968, 9.230108436653988707025818240368, 10.17113934749255520103882343439, 11.34844228280382025271017306334, 13.38436079328816462093248732137, 13.75934804257419101670095542748, 15.01961155205707779990960692353
