| L(s) = 1 | + (−0.997 + 0.0713i)2-s + (−0.212 + 0.977i)3-s + (0.989 − 0.142i)4-s + (−1.95 + 1.08i)5-s + (0.142 − 0.989i)6-s + (1.32 − 2.42i)7-s + (−0.977 + 0.212i)8-s + (−0.909 − 0.415i)9-s + (1.87 − 1.22i)10-s + (1.06 − 0.919i)11-s + (−0.0713 + 0.997i)12-s + (−0.373 + 0.203i)13-s + (−1.14 + 2.50i)14-s + (−0.649 − 2.13i)15-s + (0.959 − 0.281i)16-s + (3.65 + 4.88i)17-s + ⋯ |
| L(s) = 1 | + (−0.705 + 0.0504i)2-s + (−0.122 + 0.564i)3-s + (0.494 − 0.0711i)4-s + (−0.873 + 0.487i)5-s + (0.0580 − 0.404i)6-s + (0.499 − 0.915i)7-s + (−0.345 + 0.0751i)8-s + (−0.303 − 0.138i)9-s + (0.591 − 0.387i)10-s + (0.320 − 0.277i)11-s + (−0.0205 + 0.287i)12-s + (−0.103 + 0.0565i)13-s + (−0.306 + 0.670i)14-s + (−0.167 − 0.552i)15-s + (0.239 − 0.0704i)16-s + (0.886 + 1.18i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0260i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0260i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.913175 + 0.0119162i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.913175 + 0.0119162i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.95 - 1.08i)T \) |
| 23 | \( 1 + (2.54 + 4.06i)T \) |
| good | 7 | \( 1 + (-1.32 + 2.42i)T + (-3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (-1.06 + 0.919i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.373 - 0.203i)T + (7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (-3.65 - 4.88i)T + (-4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (1.15 + 8.04i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-5.63 - 0.809i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-5.70 - 3.66i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-1.21 - 3.26i)T + (-27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (1.16 + 2.54i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-2.34 - 0.510i)T + (39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (-9.19 - 9.19i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.54 + 1.38i)T + (28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (-2.60 + 8.86i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-6.21 + 9.66i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-0.616 - 8.61i)T + (-66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (-3.18 + 3.67i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-5.68 - 4.25i)T + (20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (-4.39 - 1.29i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (3.65 - 1.36i)T + (62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (0.0406 - 0.0261i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (2.63 + 0.982i)T + (73.3 + 63.5i)T^{2} \) |
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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
−10.63874872148574332562963591851, −9.764192057297777203312552498657, −8.553364437036329761478997132651, −8.077125770137881614989813691923, −7.02878500940368951900741641791, −6.32835130121423763979376661061, −4.77643052775802066979490598942, −3.96904886273847077569138001013, −2.80202968398928824427189495792, −0.814773627742536590878765414658,
1.06997649943906032924162309096, 2.39735388984325146155694424704, 3.82013668043967027179182266937, 5.18776545046001856315745409449, 6.04560532816502097240212977469, 7.32082657526284370334975834520, 7.934245047214510576999993481218, 8.554862626920604673987209083447, 9.495952307678448342202373544725, 10.38338213574001550646120292233
