| L(s) = 1 | + (0.0713 − 0.997i)2-s + (−0.977 + 0.212i)3-s + (−0.989 − 0.142i)4-s + (−2.10 − 0.762i)5-s + (0.142 + 0.989i)6-s + (−2.42 + 1.32i)7-s + (−0.212 + 0.977i)8-s + (0.909 − 0.415i)9-s + (−0.910 + 2.04i)10-s + (1.06 + 0.919i)11-s + (0.997 − 0.0713i)12-s + (−0.203 + 0.373i)13-s + (1.14 + 2.50i)14-s + (2.21 + 0.298i)15-s + (0.959 + 0.281i)16-s + (4.88 + 3.65i)17-s + ⋯ |
| L(s) = 1 | + (0.0504 − 0.705i)2-s + (−0.564 + 0.122i)3-s + (−0.494 − 0.0711i)4-s + (−0.940 − 0.340i)5-s + (0.0580 + 0.404i)6-s + (−0.915 + 0.499i)7-s + (−0.0751 + 0.345i)8-s + (0.303 − 0.138i)9-s + (−0.287 + 0.645i)10-s + (0.320 + 0.277i)11-s + (0.287 − 0.0205i)12-s + (−0.0565 + 0.103i)13-s + (0.306 + 0.670i)14-s + (0.572 + 0.0769i)15-s + (0.239 + 0.0704i)16-s + (1.18 + 0.886i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.322i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 + 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.840331 - 0.139403i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.840331 - 0.139403i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0713 + 0.997i)T \) |
| 3 | \( 1 + (0.977 - 0.212i)T \) |
| 5 | \( 1 + (2.10 + 0.762i)T \) |
| 23 | \( 1 + (-4.06 - 2.54i)T \) |
| good | 7 | \( 1 + (2.42 - 1.32i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (-1.06 - 0.919i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.203 - 0.373i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (-4.88 - 3.65i)T + (4.78 + 16.3i)T^{2} \) |
| 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 | \( 1 + (-3.26 - 1.21i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (1.16 - 2.54i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (0.510 + 2.34i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (-9.19 - 9.19i)T + 47iT^{2} \) |
| 53 | \( 1 + (-1.38 - 2.54i)T + (-28.6 + 44.5i)T^{2} \) |
| 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 | \( 1 + (-8.61 - 0.616i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (-3.18 - 3.67i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (4.25 + 5.68i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (4.39 - 1.29i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (1.36 - 3.65i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (-0.0406 - 0.0261i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (0.982 + 2.63i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−10.56977310936691928223451211378, −9.511011380058367321541519354926, −9.064356797469409410212267348544, −7.85311746959209956272750770283, −6.88675867153150284970834093612, −5.77198938206815360955190671952, −4.79351619351027283973590729118, −3.82004181371492137709387424712, −2.83635717141203159537421021400, −0.928284106188629478946129824617,
0.69863177890051973231215466677, 3.24663697692998042624914108157, 3.97696642188238891103321173176, 5.24823198256647214220686234181, 6.17591964627909260822147276877, 7.05448582240020335603934745039, 7.61212729181145984808410680907, 8.572063583152485007989130911322, 9.805997557034752528761634564498, 10.35144528863797624444249569709
