| L(s) = 1 | + (0.997 + 0.0713i)2-s + (−0.212 − 0.977i)3-s + (0.989 + 0.142i)4-s + (−1.79 − 1.33i)5-s + (−0.142 − 0.989i)6-s + (2.35 + 4.31i)7-s + (0.977 + 0.212i)8-s + (−0.909 + 0.415i)9-s + (−1.69 − 1.45i)10-s + (0.567 + 0.491i)11-s + (−0.0713 − 0.997i)12-s + (5.56 + 3.03i)13-s + (2.04 + 4.47i)14-s + (−0.920 + 2.03i)15-s + (0.959 + 0.281i)16-s + (0.838 − 1.12i)17-s + ⋯ |
| L(s) = 1 | + (0.705 + 0.0504i)2-s + (−0.122 − 0.564i)3-s + (0.494 + 0.0711i)4-s + (−0.802 − 0.596i)5-s + (−0.0580 − 0.404i)6-s + (0.891 + 1.63i)7-s + (0.345 + 0.0751i)8-s + (−0.303 + 0.138i)9-s + (−0.536 − 0.460i)10-s + (0.171 + 0.148i)11-s + (−0.0205 − 0.287i)12-s + (1.54 + 0.842i)13-s + (0.546 + 1.19i)14-s + (−0.237 + 0.526i)15-s + (0.239 + 0.0704i)16-s + (0.203 − 0.271i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0131i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0131i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.25277 - 0.0148432i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.25277 - 0.0148432i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.997 - 0.0713i)T \) |
| 3 | \( 1 + (0.212 + 0.977i)T \) |
| 5 | \( 1 + (1.79 + 1.33i)T \) |
| 23 | \( 1 + (-1.94 + 4.38i)T \) |
| good | 7 | \( 1 + (-2.35 - 4.31i)T + (-3.78 + 5.88i)T^{2} \) |
| 11 | \( 1 + (-0.567 - 0.491i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-5.56 - 3.03i)T + (7.02 + 10.9i)T^{2} \) |
| 17 | \( 1 + (-0.838 + 1.12i)T + (-4.78 - 16.3i)T^{2} \) |
| 19 | \( 1 + (-0.534 + 3.71i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (6.83 - 0.983i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-6.38 + 4.10i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-1.34 + 3.61i)T + (-27.9 - 24.2i)T^{2} \) |
| 41 | \( 1 + (0.469 - 1.02i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (2.55 - 0.555i)T + (39.1 - 17.8i)T^{2} \) |
| 47 | \( 1 + (8.63 - 8.63i)T - 47iT^{2} \) |
| 53 | \( 1 + (-9.22 + 5.03i)T + (28.6 - 44.5i)T^{2} \) |
| 59 | \( 1 + (-1.73 - 5.89i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (0.0184 + 0.0286i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-0.540 + 7.55i)T + (-66.3 - 9.53i)T^{2} \) |
| 71 | \( 1 + (-2.93 - 3.38i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (7.74 - 5.79i)T + (20.5 - 70.0i)T^{2} \) |
| 79 | \( 1 + (11.4 - 3.37i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (0.379 + 0.141i)T + (62.7 + 54.3i)T^{2} \) |
| 89 | \( 1 + (-10.9 - 7.04i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (2.96 - 1.10i)T + (73.3 - 63.5i)T^{2} \) |
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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
−11.16472870192027749635433606082, −9.242762290791127314091355525980, −8.603691459998741363984172608401, −7.947190664671498939554809237260, −6.79288318122524522202392567097, −5.87266935362207642032629715046, −5.02969844474918958541329628479, −4.12833137674263335330175696793, −2.70131520008284755320105610244, −1.46415483804795729809045305682,
1.23276333878295203555517071362, 3.51687389247665453216123060164, 3.69302327443153286289313664145, 4.77075105217620910292639965582, 5.88076506759485780357211281668, 6.94336240001470793693671867965, 7.81110693483511451758098271118, 8.438914045812329524899890044759, 10.18253442479298870615215486258, 10.54278909860715971616768946447
