| L(s) = 1 | + (−0.5 + 1.53i)2-s + (−0.809 − 0.587i)3-s + (−0.5 − 0.363i)4-s + (−0.690 − 2.12i)5-s + (1.30 − 0.951i)6-s + 0.618·7-s + (−1.80 + 1.31i)8-s + (−0.618 − 1.90i)9-s + 3.61·10-s + (−1.61 + 4.97i)11-s + (0.190 + 0.587i)12-s + (0.572 + 1.76i)13-s + (−0.309 + 0.951i)14-s + (−0.690 + 2.12i)15-s + (−1.50 − 4.61i)16-s + (4.23 − 3.07i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 1.08i)2-s + (−0.467 − 0.339i)3-s + (−0.250 − 0.181i)4-s + (−0.309 − 0.951i)5-s + (0.534 − 0.388i)6-s + 0.233·7-s + (−0.639 + 0.464i)8-s + (−0.206 − 0.634i)9-s + 1.14·10-s + (−0.487 + 1.50i)11-s + (0.0551 + 0.169i)12-s + (0.158 + 0.489i)13-s + (−0.0825 + 0.254i)14-s + (−0.178 + 0.549i)15-s + (−0.375 − 1.15i)16-s + (1.02 − 0.746i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.464528 + 0.218590i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.464528 + 0.218590i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.690 + 2.12i)T \) |
| good | 2 | \( 1 + (0.5 - 1.53i)T + (-1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (0.809 + 0.587i)T + (0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 - 0.618T + 7T^{2} \) |
| 11 | \( 1 + (1.61 - 4.97i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-0.572 - 1.76i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-4.23 + 3.07i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (0.690 - 0.502i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.16 + 3.57i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-2.92 - 2.12i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.42 + 1.76i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (0.0729 + 0.224i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (0.236 + 0.726i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 4.85T + 43T^{2} \) |
| 47 | \( 1 + (0.5 + 0.363i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-2.80 - 2.04i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (3.35 + 10.3i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.69 + 8.28i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (3.85 - 2.80i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-5.35 - 3.88i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (2.78 - 8.55i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-6.54 - 4.75i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-5.04 + 3.66i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (2.76 - 8.50i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-3.11 - 2.26i)T + (29.9 + 92.2i)T^{2} \) |
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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
−17.53955993142946597923869077809, −16.71523000866343881658858406906, −15.65636323923626506163918207358, −14.54451483908674810086805552473, −12.57691585280446067468854990988, −11.77994962840411386420439821725, −9.472056313954219443054734010489, −8.059710434113124745718597171927, −6.78819721404471653212800767011, −5.09978582189619516920936475269,
3.17592678118584122041376424892, 5.91719153272041345964305495670, 8.156378553166274970781992955722, 10.25003038188384685117228914797, 10.89473160089159248954574943642, 11.79657060833100328562822762174, 13.58043509454817965384085949132, 15.13834981914976864413975097058, 16.33799247012142028777306249109, 17.85867805081922335156673103748
