| L(s) = 1 | + (−0.933 + 0.358i)2-s + (−1.55 + 1.01i)3-s + (0.743 − 0.669i)4-s + (0.808 − 2.08i)5-s + (1.09 − 1.50i)6-s + (2.61 − 0.384i)7-s + (−0.453 + 0.891i)8-s + (0.180 − 0.404i)9-s + (−0.00738 + 2.23i)10-s + (1.05 + 3.14i)11-s + (−0.480 + 1.79i)12-s + (0.269 − 1.70i)13-s + (−2.30 + 1.29i)14-s + (0.849 + 4.06i)15-s + (0.104 − 0.994i)16-s + (−3.65 − 1.40i)17-s + ⋯ |
| L(s) = 1 | + (−0.660 + 0.253i)2-s + (−0.898 + 0.583i)3-s + (0.371 − 0.334i)4-s + (0.361 − 0.932i)5-s + (0.445 − 0.612i)6-s + (0.989 − 0.145i)7-s + (−0.160 + 0.315i)8-s + (0.0600 − 0.134i)9-s + (−0.00233 + 0.707i)10-s + (0.317 + 0.948i)11-s + (−0.138 + 0.517i)12-s + (0.0748 − 0.472i)13-s + (−0.616 + 0.346i)14-s + (0.219 + 1.04i)15-s + (0.0261 − 0.248i)16-s + (−0.886 − 0.340i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0146i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.955321 + 0.00698286i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.955321 + 0.00698286i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.933 - 0.358i)T \) |
| 5 | \( 1 + (-0.808 + 2.08i)T \) |
| 7 | \( 1 + (-2.61 + 0.384i)T \) |
| 11 | \( 1 + (-1.05 - 3.14i)T \) |
| good | 3 | \( 1 + (1.55 - 1.01i)T + (1.22 - 2.74i)T^{2} \) |
| 13 | \( 1 + (-0.269 + 1.70i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (3.65 + 1.40i)T + (12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (-1.85 + 2.06i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (-0.520 + 1.94i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-6.75 - 2.19i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-4.86 + 0.510i)T + (30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-1.67 + 2.58i)T + (-15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (2.00 - 0.651i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-0.687 - 0.687i)T + 43iT^{2} \) |
| 47 | \( 1 + (-3.96 + 0.207i)T + (46.7 - 4.91i)T^{2} \) |
| 53 | \( 1 + (5.08 - 6.27i)T + (-11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (-3.08 - 3.43i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (0.102 + 0.0107i)T + (59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (0.700 + 2.61i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-11.5 - 8.37i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-12.0 - 0.630i)T + (72.6 + 7.63i)T^{2} \) |
| 79 | \( 1 + (-5.47 + 12.2i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-6.31 + 0.999i)T + (78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (4.69 + 8.13i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.78 + 0.916i)T + (92.2 + 29.9i)T^{2} \) |
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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
−10.29835928647130425207038593706, −9.521501660643537123073988424987, −8.668531001968338661935687903301, −7.908263570385825616142237491771, −6.82604229643157951302444030542, −5.79397939597732302273132089990, −4.81784474443606761136369577374, −4.54399635747345744122873249476, −2.25665891558618579333621774783, −0.864021169818831928957433509390,
1.09949413818742556231546297395, 2.30003801766485000158617193052, 3.65162277542829176593204136599, 5.16694551755424150357078389225, 6.30532178963692410818755846087, 6.61358193240053899515775921648, 7.76790668257454779962876800059, 8.556752150860431781723146899550, 9.543489791914237268503338166277, 10.57382052760696617191143617147
