L(s) = 1 | + (0.707 − 0.707i)2-s + (0.707 + 0.707i)3-s − 1.00i·4-s + (2.19 − 0.408i)5-s + 1.00·6-s + (1.64 + 1.64i)7-s + (−0.707 − 0.707i)8-s + 1.00i·9-s + (1.26 − 1.84i)10-s + 6.11i·11-s + (0.707 − 0.707i)12-s + (−0.241 − 0.241i)13-s + 2.32·14-s + (1.84 + 1.26i)15-s − 1.00·16-s + (0.327 + 0.327i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (0.408 + 0.408i)3-s − 0.500i·4-s + (0.983 − 0.182i)5-s + 0.408·6-s + (0.622 + 0.622i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (0.400 − 0.582i)10-s + 1.84i·11-s + (0.204 − 0.204i)12-s + (−0.0669 − 0.0669i)13-s + 0.622·14-s + (0.475 + 0.326i)15-s − 0.250·16-s + (0.0793 + 0.0793i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0149i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0149i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.69896 - 0.0201666i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.69896 - 0.0201666i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (-2.19 + 0.408i)T \) |
| 23 | \( 1 + (1.66 + 4.49i)T \) |
good | 7 | \( 1 + (-1.64 - 1.64i)T + 7iT^{2} \) |
| 11 | \( 1 - 6.11iT - 11T^{2} \) |
| 13 | \( 1 + (0.241 + 0.241i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.327 - 0.327i)T + 17iT^{2} \) |
| 19 | \( 1 + 0.759T + 19T^{2} \) |
| 29 | \( 1 + 7.09iT - 29T^{2} \) |
| 31 | \( 1 + 8.30T + 31T^{2} \) |
| 37 | \( 1 + (1.92 + 1.92i)T + 37iT^{2} \) |
| 41 | \( 1 - 6.75T + 41T^{2} \) |
| 43 | \( 1 + (0.144 - 0.144i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.46 + 1.46i)T - 47iT^{2} \) |
| 53 | \( 1 + (-1.45 + 1.45i)T - 53iT^{2} \) |
| 59 | \( 1 + 1.68iT - 59T^{2} \) |
| 61 | \( 1 + 8.05iT - 61T^{2} \) |
| 67 | \( 1 + (3.13 + 3.13i)T + 67iT^{2} \) |
| 71 | \( 1 - 6.24T + 71T^{2} \) |
| 73 | \( 1 + (-6.62 - 6.62i)T + 73iT^{2} \) |
| 79 | \( 1 + 3.83T + 79T^{2} \) |
| 83 | \( 1 + (2.14 - 2.14i)T - 83iT^{2} \) |
| 89 | \( 1 - 4.64T + 89T^{2} \) |
| 97 | \( 1 + (11.2 + 11.2i)T + 97iT^{2} \) |
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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
−10.34215826457343462067010110115, −9.693758440646242457744213166646, −9.069268799151571885700044829976, −7.988464080577670195072681105347, −6.81382681399499354893798743803, −5.66143318008967579258333727802, −4.89424034967589281790462050406, −4.07709093101314337509141116277, −2.40605097585882379141630197189, −1.91887709586518433604909655907,
1.40378769734152605089926593791, 2.88522347028771703272920014135, 3.85559732035412480207850259653, 5.30868122006706986860350415825, 5.92561087744827331611401093144, 6.92446347963341951531511264760, 7.74172213263063488025007543789, 8.660802370752582367012846117030, 9.353484036932780025297798091633, 10.69505197143455948801674692651