| L(s) = 1 | + (−0.142 − 0.989i)2-s + (0.0575 + 1.73i)3-s + (−0.959 + 0.281i)4-s + (−1.22 − 1.87i)5-s + (1.70 − 0.303i)6-s + (1.08 − 0.696i)7-s + (0.415 + 0.909i)8-s + (−2.99 + 0.199i)9-s + (−1.67 + 1.47i)10-s + (−0.894 + 6.22i)11-s + (−0.542 − 1.64i)12-s + (1.25 − 1.95i)13-s + (−0.844 − 0.974i)14-s + (3.17 − 2.22i)15-s + (0.841 − 0.540i)16-s + (0.0693 − 0.236i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.699i)2-s + (0.0332 + 0.999i)3-s + (−0.479 + 0.140i)4-s + (−0.546 − 0.837i)5-s + (0.696 − 0.123i)6-s + (0.409 − 0.263i)7-s + (0.146 + 0.321i)8-s + (−0.997 + 0.0663i)9-s + (−0.530 + 0.467i)10-s + (−0.269 + 1.87i)11-s + (−0.156 − 0.474i)12-s + (0.347 − 0.541i)13-s + (−0.225 − 0.260i)14-s + (0.818 − 0.574i)15-s + (0.210 − 0.135i)16-s + (0.0168 − 0.0572i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.853407 + 0.526947i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.853407 + 0.526947i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 + 0.989i)T \) |
| 3 | \( 1 + (-0.0575 - 1.73i)T \) |
| 5 | \( 1 + (1.22 + 1.87i)T \) |
| 23 | \( 1 + (0.433 - 4.77i)T \) |
| good | 7 | \( 1 + (-1.08 + 0.696i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.894 - 6.22i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-1.25 + 1.95i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-0.0693 + 0.236i)T + (-14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-1.55 - 5.28i)T + (-15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-1.83 + 6.25i)T + (-24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-3.36 - 7.35i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-0.444 - 0.512i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-2.46 - 2.13i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (3.28 - 7.19i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 0.182T + 47T^{2} \) |
| 53 | \( 1 + (-3.24 - 5.04i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (5.43 - 8.46i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-2.48 + 1.13i)T + (39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.138 - 0.964i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (7.87 - 1.13i)T + (68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (2.31 + 7.90i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (1.97 - 3.06i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-1.81 + 1.56i)T + (11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-7.11 + 15.5i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (5.26 - 6.07i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.39154923819250754850846467944, −9.927801532042319719658191471707, −9.144072907790371425416239339801, −8.119418744554651264355584498064, −7.57613907308258444891484084658, −5.74506222924804645132816900455, −4.70928907642934785884845815336, −4.29178624054594954122925449740, −3.13098459444446419633262203345, −1.51370223117745221725506033371,
0.57988729620986666351418449323, 2.53099064304721610820751899495, 3.60451538139402093159062493333, 5.14604347211664770254948010068, 6.19512906847093091797721642236, 6.73101076896921290479264084358, 7.68365164015456140705395666402, 8.440391665756009155252757595851, 8.934071732257066895101204448666, 10.52493975083609329901276734413
