| L(s) = 1 | + (−0.898 − 1.09i)2-s + (−0.342 + 0.939i)3-s + (−0.386 + 1.96i)4-s + (3.60 − 0.636i)5-s + (1.33 − 0.470i)6-s + (−0.686 + 1.18i)7-s + (2.49 − 1.34i)8-s + (−0.766 − 0.642i)9-s + (−3.93 − 3.37i)10-s + (1.75 − 1.01i)11-s + (−1.71 − 1.03i)12-s + (−0.0200 − 0.0550i)13-s + (1.91 − 0.318i)14-s + (−0.636 + 3.60i)15-s + (−3.70 − 1.51i)16-s + (−0.334 + 0.280i)17-s + ⋯ |
| L(s) = 1 | + (−0.635 − 0.772i)2-s + (−0.197 + 0.542i)3-s + (−0.193 + 0.981i)4-s + (1.61 − 0.284i)5-s + (0.544 − 0.192i)6-s + (−0.259 + 0.449i)7-s + (0.880 − 0.474i)8-s + (−0.255 − 0.214i)9-s + (−1.24 − 1.06i)10-s + (0.529 − 0.305i)11-s + (−0.494 − 0.298i)12-s + (−0.00556 − 0.0152i)13-s + (0.512 − 0.0850i)14-s + (−0.164 + 0.932i)15-s + (−0.925 − 0.378i)16-s + (−0.0810 + 0.0679i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.23490 - 0.0518426i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23490 - 0.0518426i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.898 + 1.09i)T \) |
| 3 | \( 1 + (0.342 - 0.939i)T \) |
| 19 | \( 1 + (-3.22 - 2.93i)T \) |
| good | 5 | \( 1 + (-3.60 + 0.636i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (0.686 - 1.18i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.75 + 1.01i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.0200 + 0.0550i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (0.334 - 0.280i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (1.46 - 8.31i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.82 + 2.17i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-2.08 + 3.61i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 9.42iT - 37T^{2} \) |
| 41 | \( 1 + (-4.55 - 1.65i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (3.48 - 0.615i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-3.66 - 3.07i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-9.46 - 1.66i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (0.840 + 1.00i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (9.78 + 1.72i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (5.03 - 5.99i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (2.32 + 13.1i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-4.37 - 1.59i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (5.63 + 2.05i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-6.46 - 3.73i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (14.5 - 5.28i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (5.92 - 4.97i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.88207469421502400348306448487, −9.937568395767630702156714715070, −9.455949896996077696858473009607, −8.909210039549638990692255807927, −7.61196097580037409366176325109, −6.13843142525044927864904048458, −5.45515491606723644712013773741, −4.00221523597570871932606824637, −2.71274305386052665916127135032, −1.44745008726842140699113643730,
1.19119283410011584971160386936, 2.51162020935210168008620842465, 4.73651177526217297451162356831, 5.76632269618089691995914905684, 6.66135876491740703366329633285, 7.01058424278678742320159623015, 8.436236324720041395994040129653, 9.260337108509979375165133837693, 10.14259507297686839500440729705, 10.60741909190381365764824308245
