L(s) = 1 | + (0.0713 + 0.997i)2-s + (−0.977 − 0.212i)3-s + (−0.989 + 0.142i)4-s + (0.517 − 2.17i)5-s + (0.142 − 0.989i)6-s + (1.19 + 0.653i)7-s + (−0.212 − 0.977i)8-s + (0.909 + 0.415i)9-s + (2.20 + 0.361i)10-s + (−1.60 + 1.38i)11-s + (0.997 + 0.0713i)12-s + (−2.70 − 4.94i)13-s + (−0.566 + 1.24i)14-s + (−0.968 + 2.01i)15-s + (0.959 − 0.281i)16-s + (−1.68 + 1.26i)17-s + ⋯ |
L(s) = 1 | + (0.0504 + 0.705i)2-s + (−0.564 − 0.122i)3-s + (−0.494 + 0.0711i)4-s + (0.231 − 0.972i)5-s + (0.0580 − 0.404i)6-s + (0.452 + 0.247i)7-s + (−0.0751 − 0.345i)8-s + (0.303 + 0.138i)9-s + (0.697 + 0.114i)10-s + (−0.482 + 0.418i)11-s + (0.287 + 0.0205i)12-s + (−0.749 − 1.37i)13-s + (−0.151 + 0.331i)14-s + (−0.250 + 0.520i)15-s + (0.239 − 0.0704i)16-s + (−0.409 + 0.306i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.695711 - 0.517317i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.695711 - 0.517317i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.0713 - 0.997i)T \) |
| 3 | \( 1 + (0.977 + 0.212i)T \) |
| 5 | \( 1 + (-0.517 + 2.17i)T \) |
| 23 | \( 1 + (-2.15 + 4.28i)T \) |
good | 7 | \( 1 + (-1.19 - 0.653i)T + (3.78 + 5.88i)T^{2} \) |
| 11 | \( 1 + (1.60 - 1.38i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (2.70 + 4.94i)T + (-7.02 + 10.9i)T^{2} \) |
| 17 | \( 1 + (1.68 - 1.26i)T + (4.78 - 16.3i)T^{2} \) |
| 19 | \( 1 + (0.299 + 2.08i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (5.11 + 0.735i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (0.875 + 0.562i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-8.19 + 3.05i)T + (27.9 - 24.2i)T^{2} \) |
| 41 | \( 1 + (3.31 + 7.25i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-1.30 + 6.00i)T + (-39.1 - 17.8i)T^{2} \) |
| 47 | \( 1 + (1.39 - 1.39i)T - 47iT^{2} \) |
| 53 | \( 1 + (-1.46 + 2.68i)T + (-28.6 - 44.5i)T^{2} \) |
| 59 | \( 1 + (-0.317 + 1.08i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-0.435 + 0.678i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-16.0 + 1.14i)T + (66.3 - 9.53i)T^{2} \) |
| 71 | \( 1 + (7.09 - 8.18i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (0.369 - 0.494i)T + (-20.5 - 70.0i)T^{2} \) |
| 79 | \( 1 + (-13.0 - 3.84i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (0.520 + 1.39i)T + (-62.7 + 54.3i)T^{2} \) |
| 89 | \( 1 + (14.7 - 9.49i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-3.94 + 10.5i)T + (-73.3 - 63.5i)T^{2} \) |
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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
−10.20193556536940664963496289682, −9.339559099652221981535655346874, −8.380446461050734760492587551367, −7.72890156457098724588859137388, −6.73884259379774730334614921797, −5.46162821972191311152682630492, −5.22579904472719144062868878564, −4.17391098688286565465833964179, −2.28344504242051737352698236936, −0.47845321707385018174697757485,
1.69905883163720213395052703936, 2.90902330598129421696696820267, 4.12513140548501845182072732509, 5.06836565495906254090860124962, 6.14420920723388766193769614238, 7.08876633127494185934238309058, 7.970981520797734761453226058039, 9.391880473981011224080720508776, 9.827448261588672375971327244445, 10.97605640470905134829709066876