| L(s) = 1 | + (1.91 + 0.645i)2-s + (0.856 + 0.515i)3-s + (1.66 + 1.26i)4-s + (0.0203 − 0.0509i)5-s + (1.30 + 1.54i)6-s + (−3.48 + 1.61i)7-s + (0.0998 + 0.147i)8-s + (0.468 + 0.883i)9-s + (0.0718 − 0.0845i)10-s + (0.695 − 2.50i)11-s + (0.772 + 1.93i)12-s + (2.54 − 4.80i)13-s + (−7.71 + 0.839i)14-s + (0.0437 − 0.0332i)15-s + (−1.02 − 3.67i)16-s + (0.787 + 0.364i)17-s + ⋯ |
| L(s) = 1 | + (1.35 + 0.456i)2-s + (0.494 + 0.297i)3-s + (0.831 + 0.631i)4-s + (0.00908 − 0.0228i)5-s + (0.534 + 0.629i)6-s + (−1.31 + 0.609i)7-s + (0.0353 + 0.0520i)8-s + (0.156 + 0.294i)9-s + (0.0227 − 0.0267i)10-s + (0.209 − 0.754i)11-s + (0.223 + 0.559i)12-s + (0.705 − 1.33i)13-s + (−2.06 + 0.224i)14-s + (0.0112 − 0.00857i)15-s + (−0.255 − 0.919i)16-s + (0.190 + 0.0883i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.728 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.10643 + 0.835088i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.10643 + 0.835088i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.856 - 0.515i)T \) |
| 59 | \( 1 + (-3.48 - 6.84i)T \) |
| good | 2 | \( 1 + (-1.91 - 0.645i)T + (1.59 + 1.21i)T^{2} \) |
| 5 | \( 1 + (-0.0203 + 0.0509i)T + (-3.62 - 3.43i)T^{2} \) |
| 7 | \( 1 + (3.48 - 1.61i)T + (4.53 - 5.33i)T^{2} \) |
| 11 | \( 1 + (-0.695 + 2.50i)T + (-9.42 - 5.67i)T^{2} \) |
| 13 | \( 1 + (-2.54 + 4.80i)T + (-7.29 - 10.7i)T^{2} \) |
| 17 | \( 1 + (-0.787 - 0.364i)T + (11.0 + 12.9i)T^{2} \) |
| 19 | \( 1 + (2.36 - 0.521i)T + (17.2 - 7.97i)T^{2} \) |
| 23 | \( 1 + (1.35 - 8.26i)T + (-21.7 - 7.34i)T^{2} \) |
| 29 | \( 1 + (6.21 - 2.09i)T + (23.0 - 17.5i)T^{2} \) |
| 31 | \( 1 + (0.834 + 0.183i)T + (28.1 + 13.0i)T^{2} \) |
| 37 | \( 1 + (-0.774 + 1.14i)T + (-13.6 - 34.3i)T^{2} \) |
| 41 | \( 1 + (-0.402 - 2.45i)T + (-38.8 + 13.0i)T^{2} \) |
| 43 | \( 1 + (2.51 + 9.04i)T + (-36.8 + 22.1i)T^{2} \) |
| 47 | \( 1 + (-2.80 - 7.03i)T + (-34.1 + 32.3i)T^{2} \) |
| 53 | \( 1 + (-1.40 - 1.65i)T + (-8.57 + 52.3i)T^{2} \) |
| 61 | \( 1 + (0.490 + 0.165i)T + (48.5 + 36.9i)T^{2} \) |
| 67 | \( 1 + (5.10 + 7.52i)T + (-24.7 + 62.2i)T^{2} \) |
| 71 | \( 1 + (-1.88 - 4.73i)T + (-51.5 + 48.8i)T^{2} \) |
| 73 | \( 1 + (-10.4 + 1.13i)T + (71.2 - 15.6i)T^{2} \) |
| 79 | \( 1 + (6.49 - 3.90i)T + (37.0 - 69.7i)T^{2} \) |
| 83 | \( 1 + (0.626 + 11.5i)T + (-82.5 + 8.97i)T^{2} \) |
| 89 | \( 1 + (-4.10 + 1.38i)T + (70.8 - 53.8i)T^{2} \) |
| 97 | \( 1 + (-2.30 - 0.250i)T + (94.7 + 20.8i)T^{2} \) |
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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
−13.12182619788673081199276620620, −12.33019919228967646839433882337, −10.99854377058667173596991119723, −9.702797631721135501634204495701, −8.795167414754862521970314667614, −7.37521171471646079453576460816, −6.02774371589379939085526847390, −5.48634054710369855959453109633, −3.67302934431098537755664320611, −3.13354806422241985612798903108,
2.29943450012116129686406664593, 3.67530821034905101215017946883, 4.46736014838091750860551400072, 6.30433650340248469613703746545, 6.85930394941455797180204858733, 8.594353345225531897137251764829, 9.694201141594042329124255581651, 10.82910323038894322521491390002, 12.02369735312799049483251548331, 12.78910104267209680508449785243
