| L(s) = 1 | + (1.58 − 0.707i)3-s − 3.16·5-s + 1.41i·7-s + (2.00 − 2.23i)9-s − 3.16·11-s − 4.24i·13-s + (−5.00 + 2.23i)15-s − 6.70i·17-s − 3·19-s + (1.00 + 2.23i)21-s − 6.70i·23-s + 5.00·25-s + (1.58 − 4.94i)27-s + 2.23i·29-s + 4.24i·31-s + ⋯ |
| L(s) = 1 | + (0.912 − 0.408i)3-s − 1.41·5-s + 0.534i·7-s + (0.666 − 0.745i)9-s − 0.953·11-s − 1.17i·13-s + (−1.29 + 0.577i)15-s − 1.62i·17-s − 0.688·19-s + (0.218 + 0.487i)21-s − 1.39i·23-s + 1.00·25-s + (0.304 − 0.952i)27-s + 0.415i·29-s + 0.762i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.569 + 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.569 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.479938 - 0.915806i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.479938 - 0.915806i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.58 + 0.707i)T \) |
| 67 | \( 1 + (7 - 4.24i)T \) |
| good | 5 | \( 1 + 3.16T + 5T^{2} \) |
| 7 | \( 1 - 1.41iT - 7T^{2} \) |
| 11 | \( 1 + 3.16T + 11T^{2} \) |
| 13 | \( 1 + 4.24iT - 13T^{2} \) |
| 17 | \( 1 + 6.70iT - 17T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 23 | \( 1 + 6.70iT - 23T^{2} \) |
| 29 | \( 1 - 2.23iT - 29T^{2} \) |
| 31 | \( 1 - 4.24iT - 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 + 12.6T + 41T^{2} \) |
| 43 | \( 1 + 11.3iT - 43T^{2} \) |
| 47 | \( 1 - 2.23iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 2.23iT - 59T^{2} \) |
| 61 | \( 1 - 2.82iT - 61T^{2} \) |
| 71 | \( 1 - 13.4iT - 71T^{2} \) |
| 73 | \( 1 - 5T + 73T^{2} \) |
| 79 | \( 1 - 14.1iT - 79T^{2} \) |
| 83 | \( 1 + 4.47iT - 83T^{2} \) |
| 89 | \( 1 + 2.23iT - 89T^{2} \) |
| 97 | \( 1 + 8.48iT - 97T^{2} \) |
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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
−9.978351469892347463343713549106, −8.616878376809080674107488731152, −8.435100683883083931434271068388, −7.46985906217230104649218205236, −6.89916270789847921638707219963, −5.39442805229894832653822403570, −4.38563512973051706941597439941, −3.19376664674168012437709767066, −2.54219591982894105706580875697, −0.44631356795939100241948177450,
1.89843080662219645565068105752, 3.39686480807023915682682180802, 4.03755883383147708236504169715, 4.77324740785394044542911710850, 6.36067843233360608287864755165, 7.56457466248675989732278274294, 7.908024280113900994066628508722, 8.693406440329866169819453156618, 9.691594017124287663749422174347, 10.56230427035530359621520271541
