| L(s) = 1 | + (0.766 + 0.642i)2-s + (1.76 − 0.642i)3-s + (0.173 + 0.984i)4-s + (0.347 − 1.96i)5-s + (1.76 + 0.642i)6-s + (−2.53 − 4.38i)7-s + (−0.500 + 0.866i)8-s + (0.407 − 0.342i)9-s + (1.53 − 1.28i)10-s + (0.705 − 1.22i)11-s + (0.939 + 1.62i)12-s + (1.22 + 0.446i)13-s + (0.879 − 4.98i)14-s + (−0.652 − 3.70i)15-s + (−0.939 + 0.342i)16-s + (1.83 + 1.53i)17-s + ⋯ |
| L(s) = 1 | + (0.541 + 0.454i)2-s + (1.01 − 0.371i)3-s + (0.0868 + 0.492i)4-s + (0.155 − 0.880i)5-s + (0.720 + 0.262i)6-s + (−0.957 − 1.65i)7-s + (−0.176 + 0.306i)8-s + (0.135 − 0.114i)9-s + (0.484 − 0.406i)10-s + (0.212 − 0.368i)11-s + (0.271 + 0.469i)12-s + (0.340 + 0.123i)13-s + (0.235 − 1.33i)14-s + (−0.168 − 0.955i)15-s + (−0.234 + 0.0855i)16-s + (0.443 + 0.372i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.659 + 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.34207 - 1.06019i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.34207 - 1.06019i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 - 0.642i)T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + (-1.76 + 0.642i)T + (2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (-0.347 + 1.96i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (2.53 + 4.38i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.705 + 1.22i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.22 - 0.446i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.83 - 1.53i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (0.532 + 3.01i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-6.47 + 5.43i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.184 + 0.320i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4.82T + 37T^{2} \) |
| 41 | \( 1 + (-1.43 + 0.524i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.131 - 0.747i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (7.82 - 6.56i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.290 - 1.64i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-0.549 - 0.460i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.69 - 9.61i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (1.07 - 0.902i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (1.10 - 6.27i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-4.28 + 1.55i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-2.10 + 0.766i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-1.99 - 3.45i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-10.0 - 3.64i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (1.17 + 0.984i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.12623846806534679544050036296, −9.262216866347166861266633428183, −8.354547714839117227699925007217, −7.75955159481234789130741792085, −6.80303494101963607037684366384, −6.00323858585704919817513284587, −4.59130075219019827228426576860, −3.80000160683107462266079690380, −2.84160017052854163810329503593, −1.05951074331319831474131295821,
2.20093801501891534116050946743, 3.02675647567413371714970919606, 3.48438479045925389818145268374, 5.04665637047388816618815970591, 6.08557776908029923045361480777, 6.76035486327489628795232221019, 8.174402691130990572783705501518, 9.076078577094769758472740214789, 9.619774481765789166771202034639, 10.34902103316970813276645276325
