| L(s) = 1 | + (−0.149 + 1.17i)2-s + (−0.759 − 0.649i)3-s + (0.586 + 0.152i)4-s + (−0.239 − 0.0340i)5-s + (0.875 − 0.792i)6-s + (−2.71 − 3.67i)7-s + (−1.14 + 2.85i)8-s + (0.155 + 0.987i)9-s + (0.0757 − 0.275i)10-s + (0.357 + 0.115i)11-s + (−0.346 − 0.497i)12-s + (1.52 + 0.539i)13-s + (4.71 − 2.63i)14-s + (0.159 + 0.181i)15-s + (−2.11 − 1.18i)16-s + (−0.331 − 1.53i)17-s + ⋯ |
| L(s) = 1 | + (−0.106 + 0.828i)2-s + (−0.438 − 0.375i)3-s + (0.293 + 0.0763i)4-s + (−0.106 − 0.0152i)5-s + (0.357 − 0.323i)6-s + (−1.02 − 1.38i)7-s + (−0.405 + 1.00i)8-s + (0.0516 + 0.329i)9-s + (0.0239 − 0.0869i)10-s + (0.107 + 0.0346i)11-s + (−0.100 − 0.143i)12-s + (0.423 + 0.149i)13-s + (1.25 − 0.703i)14-s + (0.0412 + 0.0468i)15-s + (−0.528 − 0.295i)16-s + (−0.0803 − 0.372i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 669 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 669 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.01584 - 0.320370i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01584 - 0.320370i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.759 + 0.649i)T \) |
| 223 | \( 1 + (-14.9 + 0.870i)T \) |
| good | 2 | \( 1 + (0.149 - 1.17i)T + (-1.93 - 0.503i)T^{2} \) |
| 5 | \( 1 + (0.239 + 0.0340i)T + (4.80 + 1.39i)T^{2} \) |
| 7 | \( 1 + (2.71 + 3.67i)T + (-2.04 + 6.69i)T^{2} \) |
| 11 | \( 1 + (-0.357 - 0.115i)T + (8.93 + 6.41i)T^{2} \) |
| 13 | \( 1 + (-1.52 - 0.539i)T + (10.1 + 8.16i)T^{2} \) |
| 17 | \( 1 + (0.331 + 1.53i)T + (-15.4 + 7.00i)T^{2} \) |
| 19 | \( 1 + (-5.51 + 5.91i)T + (-1.34 - 18.9i)T^{2} \) |
| 23 | \( 1 + (-1.89 + 2.41i)T + (-5.47 - 22.3i)T^{2} \) |
| 29 | \( 1 + (-7.65 + 4.56i)T + (13.7 - 25.5i)T^{2} \) |
| 31 | \( 1 + (0.574 + 5.78i)T + (-30.3 + 6.10i)T^{2} \) |
| 37 | \( 1 + (2.32 + 2.10i)T + (3.65 + 36.8i)T^{2} \) |
| 41 | \( 1 + (2.76 - 0.474i)T + (38.6 - 13.6i)T^{2} \) |
| 43 | \( 1 + (-1.20 + 4.90i)T + (-38.1 - 19.9i)T^{2} \) |
| 47 | \( 1 + (5.41 + 5.81i)T + (-3.32 + 46.8i)T^{2} \) |
| 53 | \( 1 + (-1.66 - 0.0945i)T + (52.6 + 5.98i)T^{2} \) |
| 59 | \( 1 + (-7.97 - 0.678i)T + (58.1 + 9.97i)T^{2} \) |
| 61 | \( 1 + (0.663 + 3.56i)T + (-56.9 + 21.9i)T^{2} \) |
| 67 | \( 1 + (-3.33 - 12.1i)T + (-57.5 + 34.3i)T^{2} \) |
| 71 | \( 1 + (-6.74 - 14.3i)T + (-45.3 + 54.6i)T^{2} \) |
| 73 | \( 1 + (1.94 - 0.944i)T + (45.0 - 57.4i)T^{2} \) |
| 79 | \( 1 + (-2.48 + 6.73i)T + (-60.0 - 51.3i)T^{2} \) |
| 83 | \( 1 + (1.10 + 4.49i)T + (-73.5 + 38.4i)T^{2} \) |
| 89 | \( 1 + (0.0419 - 0.00597i)T + (85.4 - 24.8i)T^{2} \) |
| 97 | \( 1 + (1.63 + 2.08i)T + (-23.1 + 94.2i)T^{2} \) |
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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
−10.40393376789959779287733788924, −9.589309203011955289873530249720, −8.424969842896399203818932061975, −7.42696921272140384689075795133, −6.91052138735110655143939995872, −6.32413244464582139883005654578, −5.20185413311785047636163210037, −3.93798295808128965856115439699, −2.65190276476909266219921949605, −0.63894602709416142937628860875,
1.50346566147619027416281098987, 3.02354692396945638183494887355, 3.56288664025538434584921008306, 5.27719594027932237507415933662, 6.06492069082266492394852258188, 6.78191367523446466476999019416, 8.207545316404359057257611133569, 9.301569786506226302528170756774, 9.797456387218910772413932256601, 10.59970373837855472841023854701
