| L(s) = 1 | + (1.10 + 0.562i)2-s + (2.51 − 0.397i)3-s + (−0.274 − 0.377i)4-s + (1.89 − 1.18i)5-s + (2.99 + 0.973i)6-s + (−0.542 + 3.42i)7-s + (−0.477 − 3.01i)8-s + (3.30 − 1.07i)9-s + (2.75 − 0.235i)10-s + (−0.839 − 0.839i)12-s + (−0.335 + 0.658i)13-s + (−2.52 + 3.47i)14-s + (4.30 − 3.72i)15-s + (0.880 − 2.70i)16-s + (0.280 + 0.550i)17-s + (4.24 + 0.672i)18-s + ⋯ |
| L(s) = 1 | + (0.780 + 0.397i)2-s + (1.45 − 0.229i)3-s + (−0.137 − 0.188i)4-s + (0.849 − 0.528i)5-s + (1.22 + 0.397i)6-s + (−0.205 + 1.29i)7-s + (−0.168 − 1.06i)8-s + (1.10 − 0.357i)9-s + (0.872 − 0.0744i)10-s + (−0.242 − 0.242i)12-s + (−0.0931 + 0.182i)13-s + (−0.674 + 0.928i)14-s + (1.11 − 0.961i)15-s + (0.220 − 0.677i)16-s + (0.0679 + 0.133i)17-s + (1.00 + 0.158i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0313i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.43756 - 0.0538913i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.43756 - 0.0538913i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.89 + 1.18i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (-1.10 - 0.562i)T + (1.17 + 1.61i)T^{2} \) |
| 3 | \( 1 + (-2.51 + 0.397i)T + (2.85 - 0.927i)T^{2} \) |
| 7 | \( 1 + (0.542 - 3.42i)T + (-6.65 - 2.16i)T^{2} \) |
| 13 | \( 1 + (0.335 - 0.658i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.280 - 0.550i)T + (-9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (2.54 + 1.84i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (4.30 - 4.30i)T - 23iT^{2} \) |
| 29 | \( 1 + (-2.34 + 1.70i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.937 - 2.88i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (2.27 + 0.359i)T + (35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (-0.599 + 0.825i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (4.07 + 4.07i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.169 + 1.07i)T + (-44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (-3.82 - 1.94i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-4.35 - 5.98i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-3.56 - 1.15i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (9.39 + 9.39i)T + 67iT^{2} \) |
| 71 | \( 1 + (-1.11 + 3.43i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (0.531 + 0.0841i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-3.77 - 11.6i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (14.2 - 7.27i)T + (48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 - 14.6iT - 89T^{2} \) |
| 97 | \( 1 + (-6.51 + 12.7i)T + (-57.0 - 78.4i)T^{2} \) |
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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
−10.24980328526831728156643644103, −9.471183405052574171123860781998, −8.931043170363602735859157328697, −8.249400019483282055287695802663, −6.90872772165461502033607749765, −5.94637854948984718269027703973, −5.22340713867607854708172628733, −4.03341932089322582136541693453, −2.80220264292329467836609433809, −1.80558754276873975954125609482,
2.05442327348319238934275676230, 3.01217338879701235751611712352, 3.79850043988649279623725215872, 4.62819176934825597234066366974, 6.06933150213487467056682922558, 7.22060224896715420763838583681, 8.118542656452687026751634904955, 8.875968937771194695245819188449, 10.01046200556419204233428250440, 10.36358828191724967208871114670
