| L(s) = 1 | + (0.959 − 0.281i)2-s + (0.654 + 0.755i)3-s + (0.841 − 0.540i)4-s + (−0.142 − 0.989i)5-s + (0.841 + 0.540i)6-s + (1.22 − 2.68i)7-s + (0.654 − 0.755i)8-s + (−0.142 + 0.989i)9-s + (−0.415 − 0.909i)10-s + (−0.332 − 0.0977i)11-s + (0.959 + 0.281i)12-s + (−2.39 − 5.23i)13-s + (0.420 − 2.92i)14-s + (0.654 − 0.755i)15-s + (0.415 − 0.909i)16-s + (3.39 + 2.18i)17-s + ⋯ |
| L(s) = 1 | + (0.678 − 0.199i)2-s + (0.378 + 0.436i)3-s + (0.420 − 0.270i)4-s + (−0.0636 − 0.442i)5-s + (0.343 + 0.220i)6-s + (0.463 − 1.01i)7-s + (0.231 − 0.267i)8-s + (−0.0474 + 0.329i)9-s + (−0.131 − 0.287i)10-s + (−0.100 − 0.0294i)11-s + (0.276 + 0.0813i)12-s + (−0.663 − 1.45i)13-s + (0.112 − 0.781i)14-s + (0.169 − 0.195i)15-s + (0.103 − 0.227i)16-s + (0.824 + 0.529i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.670 + 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.670 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.35433 - 1.04588i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.35433 - 1.04588i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.959 + 0.281i)T \) |
| 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 5 | \( 1 + (0.142 + 0.989i)T \) |
| 23 | \( 1 + (-4.27 + 2.18i)T \) |
| good | 7 | \( 1 + (-1.22 + 2.68i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (0.332 + 0.0977i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (2.39 + 5.23i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-3.39 - 2.18i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-0.353 + 0.227i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-3.21 - 2.06i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (6.38 - 7.36i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (1.02 - 7.10i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.0904 - 0.629i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (0.142 + 0.163i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 5.27T + 47T^{2} \) |
| 53 | \( 1 + (-1.84 + 4.03i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (3.02 + 6.62i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (0.141 - 0.163i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (1.64 - 0.483i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-3.18 + 0.934i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (3.03 - 1.94i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-4.64 - 10.1i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (2.32 - 16.2i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (0.189 + 0.219i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-1.46 - 10.1i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.50412249945241063542897290864, −9.755909283902645794947294566119, −8.502524714012801295974437181848, −7.77473545229251554321593973011, −6.88639347006563123208941822201, −5.39835571487870623821478683682, −4.88824088849768466451214156270, −3.78012638193845432103219210454, −2.89650612028051066195385277766, −1.17656422085549305404647491215,
1.95745636574292369216862541748, 2.81248494058482183573680888961, 4.08429386597544961739569276102, 5.22593602668844020167744406491, 6.06304206035086787697490219464, 7.18055863084023139407825068167, 7.64596262252487661047603030224, 8.896957846200771586652670198004, 9.483243091312004119701094706953, 10.81913521372132403156290390615
