L(s) = 1 | + (1.35 − 0.780i)2-s + (0.866 − 0.5i)3-s + (0.219 − 0.379i)4-s + (0.780 − 1.35i)6-s + (0.486 + 0.280i)7-s + 2.43i·8-s + (0.499 − 0.866i)9-s + (1 + 1.73i)11-s − 0.438i·12-s + (3.57 + 0.5i)13-s + 0.876·14-s + (2.34 + 4.05i)16-s + (−1.35 − 0.780i)17-s − 1.56i·18-s + (3.56 − 6.16i)19-s + ⋯ |
L(s) = 1 | + (0.956 − 0.552i)2-s + (0.499 − 0.288i)3-s + (0.109 − 0.189i)4-s + (0.318 − 0.552i)6-s + (0.183 + 0.106i)7-s + 0.862i·8-s + (0.166 − 0.288i)9-s + (0.301 + 0.522i)11-s − 0.126i·12-s + (0.990 + 0.138i)13-s + 0.234·14-s + (0.585 + 1.01i)16-s + (−0.327 − 0.189i)17-s − 0.368i·18-s + (0.817 − 1.41i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 + 0.438i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.898 + 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.13841 - 0.725678i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.13841 - 0.725678i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-3.57 - 0.5i)T \) |
good | 2 | \( 1 + (-1.35 + 0.780i)T + (1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (-0.486 - 0.280i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.35 + 0.780i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.56 + 6.16i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.73 + i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.34 - 5.78i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 2.56T + 31T^{2} \) |
| 37 | \( 1 + (6.54 - 3.78i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.780 - 1.35i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.95 + 2.28i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 8.24iT - 47T^{2} \) |
| 53 | \( 1 + 0.684iT - 53T^{2} \) |
| 59 | \( 1 + (1.43 - 2.49i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.93 - 3.35i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.95 - 2.28i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (7 - 12.1i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 10.1iT - 73T^{2} \) |
| 79 | \( 1 + 5.43T + 79T^{2} \) |
| 83 | \( 1 + 0.876iT - 83T^{2} \) |
| 89 | \( 1 + (-2.43 - 4.22i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.41 + 4.28i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.06732286783213999963296436428, −8.816092177192722325412765729458, −8.588230699781054144205738631802, −7.29480951430238469089032474483, −6.55940546557949332194826632655, −5.26458535143657212794763362110, −4.56525181134168341328668642347, −3.50599444400209957086459850098, −2.72500034085502159024755600444, −1.51407951535583675676101384000,
1.33985340787563245358419579091, 3.17067801031100306724044941323, 3.87298609046546822008198798965, 4.75167829021221585162621151891, 5.81264077071272982110643473102, 6.34163497248020114133541158221, 7.51406179464266287343580907852, 8.285295537076272923493546038851, 9.210720662330515409859294402945, 10.02732546837187531563937661435