| 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
−10.97605640470905134829709066876, −9.827448261588672375971327244445, −9.391880473981011224080720508776, −7.970981520797734761453226058039, −7.08876633127494185934238309058, −6.14420920723388766193769614238, −5.06836565495906254090860124962, −4.12513140548501845182072732509, −2.90902330598129421696696820267, −1.69905883163720213395052703936,
0.47845321707385018174697757485, 2.28344504242051737352698236936, 4.17391098688286565465833964179, 5.22579904472719144062868878564, 5.46162821972191311152682630492, 6.73884259379774730334614921797, 7.72890156457098724588859137388, 8.380446461050734760492587551367, 9.339559099652221981535655346874, 10.20193556536940664963496289682
