| L(s) = 1 | + (−0.380 − 1.86i)2-s + (0.987 + 0.160i)3-s + (−1.49 + 0.637i)4-s + (1.63 − 0.403i)5-s + (−0.0766 − 1.90i)6-s + (0.579 − 1.22i)7-s + (−0.404 − 0.586i)8-s + (0.948 + 0.316i)9-s + (−1.37 − 2.90i)10-s + (5.06 − 1.68i)11-s + (−1.57 + 0.389i)12-s + (1.76 + 3.14i)13-s + (−2.49 − 0.615i)14-s + (1.68 − 0.135i)15-s + (−3.19 + 3.32i)16-s + (−2.80 + 5.90i)17-s + ⋯ |
| L(s) = 1 | + (−0.269 − 1.31i)2-s + (0.569 + 0.0926i)3-s + (−0.747 + 0.318i)4-s + (0.732 − 0.180i)5-s + (−0.0313 − 0.776i)6-s + (0.218 − 0.461i)7-s + (−0.143 − 0.207i)8-s + (0.316 + 0.105i)9-s + (−0.435 − 0.917i)10-s + (1.52 − 0.509i)11-s + (−0.455 + 0.112i)12-s + (0.489 + 0.871i)13-s + (−0.667 − 0.164i)14-s + (0.434 − 0.0350i)15-s + (−0.798 + 0.830i)16-s + (−0.680 + 1.43i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.410 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.983661 - 1.52153i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.983661 - 1.52153i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.987 - 0.160i)T \) |
| 13 | \( 1 + (-1.76 - 3.14i)T \) |
| good | 2 | \( 1 + (0.380 + 1.86i)T + (-1.83 + 0.783i)T^{2} \) |
| 5 | \( 1 + (-1.63 + 0.403i)T + (4.42 - 2.32i)T^{2} \) |
| 7 | \( 1 + (-0.579 + 1.22i)T + (-4.42 - 5.42i)T^{2} \) |
| 11 | \( 1 + (-5.06 + 1.68i)T + (8.79 - 6.60i)T^{2} \) |
| 17 | \( 1 + (2.80 - 5.90i)T + (-10.7 - 13.1i)T^{2} \) |
| 19 | \( 1 + (0.853 + 1.47i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.49 + 6.05i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.278 + 1.36i)T + (-26.6 + 11.3i)T^{2} \) |
| 31 | \( 1 + (8.00 + 4.20i)T + (17.6 + 25.5i)T^{2} \) |
| 37 | \( 1 + (-0.930 + 0.588i)T + (15.8 - 33.4i)T^{2} \) |
| 41 | \( 1 + (2.37 + 0.386i)T + (38.8 + 12.9i)T^{2} \) |
| 43 | \( 1 + (6.60 + 4.17i)T + (18.4 + 38.8i)T^{2} \) |
| 47 | \( 1 + (-0.459 - 3.78i)T + (-45.6 + 11.2i)T^{2} \) |
| 53 | \( 1 + (1.93 + 2.79i)T + (-18.7 + 49.5i)T^{2} \) |
| 59 | \( 1 + (-10.4 - 10.8i)T + (-2.37 + 58.9i)T^{2} \) |
| 61 | \( 1 + (10.4 + 0.842i)T + (60.2 + 9.78i)T^{2} \) |
| 67 | \( 1 + (5.84 + 2.48i)T + (46.4 + 48.3i)T^{2} \) |
| 71 | \( 1 + (-4.93 + 6.04i)T + (-14.2 - 69.5i)T^{2} \) |
| 73 | \( 1 + (-6.31 - 5.59i)T + (8.79 + 72.4i)T^{2} \) |
| 79 | \( 1 + (-1.17 - 9.69i)T + (-76.7 + 18.9i)T^{2} \) |
| 83 | \( 1 + (5.93 - 15.6i)T + (-62.1 - 55.0i)T^{2} \) |
| 89 | \( 1 + (-0.390 + 0.675i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.21 + 11.1i)T + (-81.9 + 51.8i)T^{2} \) |
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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
−10.79769328991816505017198441395, −9.736104336703950179803379297478, −9.027142087602230766726056432969, −8.562652621642415533119660435450, −6.86322528297129058046622616425, −6.12130465530780247549106821801, −4.25344625458737063707692566422, −3.70450628126698188999846996105, −2.17129336108419897981621999372, −1.34430307810029930649876110123,
1.85246132904364993581931503592, 3.31640839340421197737020766673, 4.94004364425873472612422460557, 5.85001263756033047016830192667, 6.78295932040146656707422243269, 7.39900425942812284191660736270, 8.541004728626812950677619439754, 9.163130467173975253935328308445, 9.777783733390602796051707027186, 11.22013702994164067696712578251
