L(s) = 1 | + (−1.12 − 1.94i)2-s + (1.56 + 0.742i)3-s + (−1.53 + 2.65i)4-s + (−0.313 − 3.88i)6-s + (2.22 − 1.42i)7-s + 2.39·8-s + (1.89 + 2.32i)9-s + (−1.64 − 0.952i)11-s + (−4.36 + 3.01i)12-s + 5.07·13-s + (−5.29 − 2.73i)14-s + (0.369 + 0.639i)16-s + (3.85 + 2.22i)17-s + (2.39 − 6.31i)18-s + (−3.85 + 2.22i)19-s + ⋯ |
L(s) = 1 | + (−0.795 − 1.37i)2-s + (0.903 + 0.428i)3-s + (−0.766 + 1.32i)4-s + (−0.128 − 1.58i)6-s + (0.841 − 0.540i)7-s + 0.846·8-s + (0.632 + 0.774i)9-s + (−0.497 − 0.287i)11-s + (−1.26 + 0.870i)12-s + 1.40·13-s + (−1.41 − 0.730i)14-s + (0.0923 + 0.159i)16-s + (0.936 + 0.540i)17-s + (0.564 − 1.48i)18-s + (−0.884 + 0.510i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.193 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.193 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.08483 - 0.892147i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08483 - 0.892147i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.56 - 0.742i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.22 + 1.42i)T \) |
good | 2 | \( 1 + (1.12 + 1.94i)T + (-1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (1.64 + 0.952i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5.07T + 13T^{2} \) |
| 17 | \( 1 + (-3.85 - 2.22i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.85 - 2.22i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.42 + 2.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 8.82iT - 29T^{2} \) |
| 31 | \( 1 + (-4.81 - 2.77i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.02 + 2.32i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 0.250T + 41T^{2} \) |
| 43 | \( 1 + 9.23iT - 43T^{2} \) |
| 47 | \( 1 + (2.66 - 1.53i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.21 + 2.09i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6.95 - 12.0i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.51 - 0.874i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7.18 + 4.14i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 9.68iT - 71T^{2} \) |
| 73 | \( 1 + (3.15 - 5.47i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.59 + 2.76i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 8.98iT - 83T^{2} \) |
| 89 | \( 1 + (-5.43 - 9.41i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 8.94T + 97T^{2} \) |
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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
−10.41307016383150427843056131413, −10.17220190015133568098961554953, −8.886180581244703225051528216500, −8.301259926830019553792974811582, −7.77330849590282434251978532734, −5.97620456637281135193485601348, −4.30886019341074706440771142436, −3.63414474063942433374871956332, −2.41270511691150181678544756807, −1.27467854138014411902532700846,
1.39064565647228836769301493374, 3.03883943148755201560460308721, 4.70320078559628340146590353578, 5.86487995824585698134487484368, 6.71087174879599129174435590109, 7.77645345309591314503830059973, 8.166823734859770875034902333883, 8.933786483325068997425459309191, 9.643231984261452820890587149412, 10.80777554333874722049279883023