| L(s) = 1 | − 1.29·2-s − 0.320·4-s + (1.01 + 1.99i)5-s + (2.26 + 1.37i)7-s + 3.00·8-s + (−1.31 − 2.58i)10-s + 3.12i·11-s + 2.45·13-s + (−2.93 − 1.77i)14-s − 3.25·16-s + 4.43i·17-s − 4.17i·19-s + (−0.324 − 0.639i)20-s − 4.04i·22-s + 5.77·23-s + ⋯ |
| L(s) = 1 | − 0.916·2-s − 0.160·4-s + (0.452 + 0.891i)5-s + (0.855 + 0.518i)7-s + 1.06·8-s + (−0.414 − 0.817i)10-s + 0.941i·11-s + 0.681·13-s + (−0.783 − 0.474i)14-s − 0.814·16-s + 1.07i·17-s − 0.957i·19-s + (−0.0725 − 0.142i)20-s − 0.862i·22-s + 1.20·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0750 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0750 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.771642 + 0.715758i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.771642 + 0.715758i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.01 - 1.99i)T \) |
| 7 | \( 1 + (-2.26 - 1.37i)T \) |
| good | 2 | \( 1 + 1.29T + 2T^{2} \) |
| 11 | \( 1 - 3.12iT - 11T^{2} \) |
| 13 | \( 1 - 2.45T + 13T^{2} \) |
| 17 | \( 1 - 4.43iT - 17T^{2} \) |
| 19 | \( 1 + 4.17iT - 19T^{2} \) |
| 23 | \( 1 - 5.77T + 23T^{2} \) |
| 29 | \( 1 - 0.339iT - 29T^{2} \) |
| 31 | \( 1 + 4.40iT - 31T^{2} \) |
| 37 | \( 1 + 11.2iT - 37T^{2} \) |
| 41 | \( 1 + 5.43T + 41T^{2} \) |
| 43 | \( 1 - 0.439iT - 43T^{2} \) |
| 47 | \( 1 - 10.2iT - 47T^{2} \) |
| 53 | \( 1 - 2.76T + 53T^{2} \) |
| 59 | \( 1 + 5.00T + 59T^{2} \) |
| 61 | \( 1 - 9.34iT - 61T^{2} \) |
| 67 | \( 1 - 15.7iT - 67T^{2} \) |
| 71 | \( 1 + 10.1iT - 71T^{2} \) |
| 73 | \( 1 + 9.26T + 73T^{2} \) |
| 79 | \( 1 - 15.6T + 79T^{2} \) |
| 83 | \( 1 - 4.13iT - 83T^{2} \) |
| 89 | \( 1 - 5.20T + 89T^{2} \) |
| 97 | \( 1 + 6.09T + 97T^{2} \) |
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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
−10.24900513206480359477709319021, −9.244970511876854978883419503054, −8.812158186147799980863622021940, −7.75495441579263860298873791240, −7.15113757036924875098577763784, −6.05423117123753709903430337763, −5.01882357072283198375016651703, −4.01728473893504909804903809860, −2.46888443336168796992720371705, −1.45987779469430007934086878161,
0.78861055395124390622033677283, 1.62753741944018646301151953534, 3.49516287382631161322087101756, 4.75951678129928878310697410032, 5.26235007394937928211771688083, 6.57432034215065135633498979168, 7.68941768841007491974654238569, 8.455655172442969113275799366221, 8.806376647862540201625719460058, 9.764355220432519818604142202093
