| L(s) = 1 | + (−1.08 − 0.910i)2-s + (0.861 + 1.28i)3-s + (0.343 + 1.97i)4-s + (1.07 + 0.213i)5-s + (0.240 − 2.17i)6-s + (0.127 + 0.641i)7-s + (1.42 − 2.44i)8-s + (0.228 − 0.552i)9-s + (−0.967 − 1.20i)10-s + (−3.00 − 2.01i)11-s + (−2.24 + 2.13i)12-s + (1.76 + 1.76i)13-s + (0.445 − 0.809i)14-s + (0.649 + 1.56i)15-s + (−3.76 + 1.35i)16-s + (−4.12 + 0.0241i)17-s + ⋯ |
| L(s) = 1 | + (−0.765 − 0.643i)2-s + (0.497 + 0.744i)3-s + (0.171 + 0.985i)4-s + (0.480 + 0.0955i)5-s + (0.0983 − 0.889i)6-s + (0.0481 + 0.242i)7-s + (0.502 − 0.864i)8-s + (0.0762 − 0.184i)9-s + (−0.306 − 0.382i)10-s + (−0.907 − 0.606i)11-s + (−0.647 + 0.617i)12-s + (0.488 + 0.488i)13-s + (0.119 − 0.216i)14-s + (0.167 + 0.404i)15-s + (−0.941 + 0.338i)16-s + (−0.999 + 0.00585i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0261i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.774287 + 0.0101204i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.774287 + 0.0101204i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.08 + 0.910i)T \) |
| 17 | \( 1 + (4.12 - 0.0241i)T \) |
| good | 3 | \( 1 + (-0.861 - 1.28i)T + (-1.14 + 2.77i)T^{2} \) |
| 5 | \( 1 + (-1.07 - 0.213i)T + (4.61 + 1.91i)T^{2} \) |
| 7 | \( 1 + (-0.127 - 0.641i)T + (-6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (3.00 + 2.01i)T + (4.20 + 10.1i)T^{2} \) |
| 13 | \( 1 + (-1.76 - 1.76i)T + 13iT^{2} \) |
| 19 | \( 1 + (5.03 - 2.08i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-5.17 + 7.75i)T + (-8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (0.213 - 1.07i)T + (-26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 + (-0.159 + 0.106i)T + (11.8 - 28.6i)T^{2} \) |
| 37 | \( 1 + (-3.44 + 2.29i)T + (14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (8.33 - 1.65i)T + (37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (-3.49 - 1.44i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (6.02 - 6.02i)T - 47iT^{2} \) |
| 53 | \( 1 + (-4.17 - 10.0i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (0.832 - 2.00i)T + (-41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (-1.06 - 5.37i)T + (-56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 - 6.44T + 67T^{2} \) |
| 71 | \( 1 + (2.02 + 3.02i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (0.183 + 0.0364i)T + (67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (-1.38 - 0.926i)T + (30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (-4.62 - 11.1i)T + (-58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (3.79 - 3.79i)T - 89iT^{2} \) |
| 97 | \( 1 + (-3.00 + 15.1i)T + (-89.6 - 37.1i)T^{2} \) |
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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
−14.95932499750837891663389005716, −13.56144796197425764253355973737, −12.53956617771181310464235406406, −11.01706506303096964000547311853, −10.28645133111847104747486692492, −9.069065434699849684672306966603, −8.385188015475125446288813314211, −6.50076571748384032037722986786, −4.25237807371468819665627261041, −2.60480072413881245021048445243,
2.03919032930974263781579135419, 5.16383863309987229900702517374, 6.74375311003277069715723290406, 7.73729371757478104583675924469, 8.767873382537452874977725459530, 10.06916995073173331854933524706, 11.12636238326849956037583854697, 13.20652509040793048873515796242, 13.46422686501704516572820739620, 15.01722640176333734248275118312
