| L(s) = 1 | + (0.292 + 1.70i)3-s + 3.41i·5-s + i·7-s + (−2.82 + i)9-s + 2·11-s + 3.41·13-s + (−5.82 + i)15-s + 4.82i·17-s − 6.24i·19-s + (−1.70 + 0.292i)21-s − 4·23-s − 6.65·25-s + (−2.53 − 4.53i)27-s + 4.82i·29-s + 1.17i·31-s + ⋯ |
| L(s) = 1 | + (0.169 + 0.985i)3-s + 1.52i·5-s + 0.377i·7-s + (−0.942 + 0.333i)9-s + 0.603·11-s + 0.946·13-s + (−1.50 + 0.258i)15-s + 1.17i·17-s − 1.43i·19-s + (−0.372 + 0.0639i)21-s − 0.834·23-s − 1.33·25-s + (−0.487 − 0.872i)27-s + 0.896i·29-s + 0.210i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.448578 + 1.41134i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.448578 + 1.41134i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.292 - 1.70i)T \) |
| 7 | \( 1 - iT \) |
| good | 5 | \( 1 - 3.41iT - 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 - 4.82iT - 17T^{2} \) |
| 19 | \( 1 + 6.24iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 4.82iT - 29T^{2} \) |
| 31 | \( 1 - 1.17iT - 31T^{2} \) |
| 37 | \( 1 + 0.828T + 37T^{2} \) |
| 41 | \( 1 + 10iT - 41T^{2} \) |
| 43 | \( 1 + 10.4iT - 43T^{2} \) |
| 47 | \( 1 - 1.65T + 47T^{2} \) |
| 53 | \( 1 - 13.3iT - 53T^{2} \) |
| 59 | \( 1 + 6.24T + 59T^{2} \) |
| 61 | \( 1 - 14.2T + 61T^{2} \) |
| 67 | \( 1 - 9.31iT - 67T^{2} \) |
| 71 | \( 1 + 0.343T + 71T^{2} \) |
| 73 | \( 1 + 3.17T + 73T^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 - 7.65iT - 89T^{2} \) |
| 97 | \( 1 - 5.31T + 97T^{2} \) |
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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
−10.73407305884765528279206925216, −10.24902351613322057564825658057, −9.083342638646280770869906167954, −8.537079843534201443580321171218, −7.23002652534497141722947204136, −6.35407543575744745154962984332, −5.53793252578151470096807336824, −4.07590662791715287407549147156, −3.39169834508120182371077619166, −2.28729350308410129217225848829,
0.810101194556645063547999786689, 1.78610389225483470104718565804, 3.52871098273680344426740647452, 4.62243800885960709008629911594, 5.78277286514227101071816200424, 6.51197269289063589164076919353, 7.85247088717976281955677758442, 8.218413224884172381167507537982, 9.179894096290314176672868295764, 9.893760986187044794616736145966
