| L(s) = 1 | + (0.209 − 2.59i)2-s + (−0.845 − 0.534i)3-s + (−4.69 − 0.762i)4-s + (−1.02 + 1.16i)5-s + (−1.56 + 2.07i)6-s + (3.12 + 3.00i)7-s + (−1.71 + 6.94i)8-s + (0.428 + 0.903i)9-s + (2.79 + 2.90i)10-s + (2.25 + 1.06i)11-s + (3.55 + 3.15i)12-s + (−3.36 + 1.30i)13-s + (8.43 − 7.46i)14-s + (1.49 − 0.431i)15-s + (8.62 + 2.87i)16-s + (−4.23 + 4.40i)17-s + ⋯ |
| L(s) = 1 | + (0.147 − 1.83i)2-s + (−0.487 − 0.308i)3-s + (−2.34 − 0.381i)4-s + (−0.460 + 0.519i)5-s + (−0.637 + 0.848i)6-s + (1.18 + 1.13i)7-s + (−0.605 + 2.45i)8-s + (0.142 + 0.301i)9-s + (0.883 + 0.919i)10-s + (0.679 + 0.322i)11-s + (1.02 + 0.909i)12-s + (−0.932 + 0.362i)13-s + (2.25 − 1.99i)14-s + (0.384 − 0.111i)15-s + (2.15 + 0.719i)16-s + (−1.02 + 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 + 0.562i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.826 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.879730 - 0.270719i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.879730 - 0.270719i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.845 + 0.534i)T \) |
| 13 | \( 1 + (3.36 - 1.30i)T \) |
| good | 2 | \( 1 + (-0.209 + 2.59i)T + (-1.97 - 0.320i)T^{2} \) |
| 5 | \( 1 + (1.02 - 1.16i)T + (-0.602 - 4.96i)T^{2} \) |
| 7 | \( 1 + (-3.12 - 3.00i)T + (0.281 + 6.99i)T^{2} \) |
| 11 | \( 1 + (-2.25 - 1.06i)T + (6.95 + 8.52i)T^{2} \) |
| 17 | \( 1 + (4.23 - 4.40i)T + (-0.684 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-4.62 - 2.66i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.02 + 3.51i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (5.74 + 0.463i)T + (28.6 + 4.65i)T^{2} \) |
| 31 | \( 1 + (-1.44 - 0.175i)T + (30.0 + 7.41i)T^{2} \) |
| 37 | \( 1 + (-1.94 - 4.56i)T + (-25.6 + 26.6i)T^{2} \) |
| 41 | \( 1 + (-4.98 + 7.88i)T + (-17.5 - 37.0i)T^{2} \) |
| 43 | \( 1 + (6.93 + 2.95i)T + (29.7 + 31.0i)T^{2} \) |
| 47 | \( 1 + (-7.98 - 3.02i)T + (35.1 + 31.1i)T^{2} \) |
| 53 | \( 1 + (-0.615 - 0.151i)T + (46.9 + 24.6i)T^{2} \) |
| 59 | \( 1 + (0.902 + 2.70i)T + (-47.1 + 35.4i)T^{2} \) |
| 61 | \( 1 + (3.18 - 11.0i)T + (-51.5 - 32.6i)T^{2} \) |
| 67 | \( 1 + (-2.47 - 15.1i)T + (-63.5 + 21.2i)T^{2} \) |
| 71 | \( 1 + (3.89 + 0.156i)T + (70.7 + 5.71i)T^{2} \) |
| 73 | \( 1 + (-8.43 + 5.82i)T + (25.8 - 68.2i)T^{2} \) |
| 79 | \( 1 + (-4.06 + 10.7i)T + (-59.1 - 52.3i)T^{2} \) |
| 83 | \( 1 + (3.28 - 6.26i)T + (-47.1 - 68.3i)T^{2} \) |
| 89 | \( 1 + (1.25 - 0.725i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.198 - 0.0405i)T + (89.2 - 38.0i)T^{2} \) |
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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
−11.11873587509047412912591368088, −10.31274679335282820153042343444, −9.272445664910629835068415152648, −8.494473149524503933756151629779, −7.34210655963390050761098736080, −5.80914551657788323911713361074, −4.79321186178667161216607287276, −3.91454873837660749329142646021, −2.43206632513196566328345058742, −1.63705750825187357372289141052,
0.59718579469611256011229672096, 3.92822722136443522619753164387, 4.74521264046465665011185093777, 5.20248740156489148087203456519, 6.51990323917743432248564032694, 7.46762043843416576359746678899, 7.83031626689721253672042673128, 9.010951304949301733248097397143, 9.704833134428771996227865462001, 11.11498848021256716675165877633
