| L(s) = 1 | + (0.342 + 0.939i)2-s + (0.939 − 0.342i)3-s + (−0.766 + 0.642i)4-s + (−1.43 + 1.71i)5-s + (0.642 + 0.766i)6-s + (2.64 + 0.161i)7-s + (−0.866 − 0.500i)8-s + (0.766 − 0.642i)9-s + (−2.10 − 0.764i)10-s + (0.719 + 1.24i)11-s + (−0.500 + 0.866i)12-s + (2.28 − 1.91i)13-s + (0.751 + 2.53i)14-s + (−0.764 + 2.10i)15-s + (0.173 − 0.984i)16-s + (−2.36 + 2.82i)17-s + ⋯ |
| L(s) = 1 | + (0.241 + 0.664i)2-s + (0.542 − 0.197i)3-s + (−0.383 + 0.321i)4-s + (−0.642 + 0.765i)5-s + (0.262 + 0.312i)6-s + (0.998 + 0.0610i)7-s + (−0.306 − 0.176i)8-s + (0.255 − 0.214i)9-s + (−0.664 − 0.241i)10-s + (0.216 + 0.375i)11-s + (−0.144 + 0.249i)12-s + (0.634 − 0.532i)13-s + (0.200 + 0.677i)14-s + (−0.197 + 0.542i)15-s + (0.0434 − 0.246i)16-s + (−0.574 + 0.684i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.172 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.172 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.26834 + 1.51003i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26834 + 1.51003i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.342 - 0.939i)T \) |
| 3 | \( 1 + (-0.939 + 0.342i)T \) |
| 7 | \( 1 + (-2.64 - 0.161i)T \) |
| 19 | \( 1 + (-2.01 - 3.86i)T \) |
| good | 5 | \( 1 + (1.43 - 1.71i)T + (-0.868 - 4.92i)T^{2} \) |
| 11 | \( 1 + (-0.719 - 1.24i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.28 + 1.91i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (2.36 - 2.82i)T + (-2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.833 - 4.72i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (1.14 - 0.202i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + 2.17T + 31T^{2} \) |
| 37 | \( 1 + (-4.92 + 2.84i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.83 + 2.38i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (8.40 - 3.05i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (1.68 + 2.00i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-3.13 - 3.73i)T + (-9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (1.35 + 1.13i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (1.66 - 0.294i)T + (57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.11 + 8.55i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (0.679 + 1.86i)T + (-54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-0.716 - 1.96i)T + (-55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-11.9 - 2.10i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-8.45 + 4.88i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.147 - 0.0536i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-1.41 + 8.01i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−10.59668160061417779392905951138, −9.460718591691674554877296640771, −8.476652083075168109313402929228, −7.82740179627087601552767474243, −7.27122159485378298516319615619, −6.25006231061629142135013724929, −5.21580196201482952861219912520, −4.01383441024901579074538073865, −3.30794523749233562123645361287, −1.72100401949950909840632507145,
0.950592295795312382875260453086, 2.31306358206260313220333683678, 3.61183678424767896203140990368, 4.55270910028739598983980285067, 5.04171025506228492629587778033, 6.56302365596089009801935661317, 7.73980445936859782268134300990, 8.665307322140668240025375051872, 8.930320098581667610270395513723, 10.06501359934465096772450945181
