L(s) = 1 | + (0.636 + 0.636i)3-s + (1.68 − 1.47i)5-s + (0.707 − 0.707i)7-s − 2.19i·9-s − 0.0126i·11-s + (1.39 − 1.39i)13-s + (2.00 + 0.134i)15-s + (−1.83 − 1.83i)17-s − 1.25·19-s + 0.899·21-s + (0.761 + 0.761i)23-s + (0.666 − 4.95i)25-s + (3.30 − 3.30i)27-s + 1.27i·29-s + 8.86i·31-s + ⋯ |
L(s) = 1 | + (0.367 + 0.367i)3-s + (0.752 − 0.658i)5-s + (0.267 − 0.267i)7-s − 0.730i·9-s − 0.00380i·11-s + (0.386 − 0.386i)13-s + (0.518 + 0.0346i)15-s + (−0.444 − 0.444i)17-s − 0.288·19-s + 0.196·21-s + (0.158 + 0.158i)23-s + (0.133 − 0.991i)25-s + (0.635 − 0.635i)27-s + 0.237i·29-s + 1.59i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 + 0.531i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.847 + 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.80884 - 0.520585i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.80884 - 0.520585i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.68 + 1.47i)T \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 3 | \( 1 + (-0.636 - 0.636i)T + 3iT^{2} \) |
| 11 | \( 1 + 0.0126iT - 11T^{2} \) |
| 13 | \( 1 + (-1.39 + 1.39i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.83 + 1.83i)T + 17iT^{2} \) |
| 19 | \( 1 + 1.25T + 19T^{2} \) |
| 23 | \( 1 + (-0.761 - 0.761i)T + 23iT^{2} \) |
| 29 | \( 1 - 1.27iT - 29T^{2} \) |
| 31 | \( 1 - 8.86iT - 31T^{2} \) |
| 37 | \( 1 + (0.649 + 0.649i)T + 37iT^{2} \) |
| 41 | \( 1 - 5.52T + 41T^{2} \) |
| 43 | \( 1 + (-4.70 - 4.70i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.07 - 2.07i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.69 - 2.69i)T - 53iT^{2} \) |
| 59 | \( 1 + 0.236T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 + (-4.86 + 4.86i)T - 67iT^{2} \) |
| 71 | \( 1 - 12.9iT - 71T^{2} \) |
| 73 | \( 1 + (4.87 - 4.87i)T - 73iT^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 + (12.2 + 12.2i)T + 83iT^{2} \) |
| 89 | \( 1 - 8.25iT - 89T^{2} \) |
| 97 | \( 1 + (-4.81 - 4.81i)T + 97iT^{2} \) |
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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
−10.52541420108377158891438393314, −9.712336493210854451754338627726, −8.954307418168549918210343272097, −8.334580388815414033304397374386, −7.02534015419240250738971578439, −6.04835937849996865284620061046, −5.01906929183222358459044170846, −4.03792442034615621578179353197, −2.76725681895926823951841254696, −1.17485923068331199593983970551,
1.83788234552901948024710560113, 2.63763019747611430131595831106, 4.13361512882420509204892511455, 5.43695057590829494181606455401, 6.32656338214804818119487271188, 7.25460838782257301295537679479, 8.186545158810851123320766679048, 9.035115580397645738856702246560, 9.997547610728693933050855655081, 10.88062346412233845717096040498