| L(s) = 1 | + (0.930 + 0.365i)2-s + (1.49 + 0.882i)3-s + (0.733 + 0.680i)4-s + (0.128 + 0.0879i)5-s + (1.06 + 1.36i)6-s + (−2.44 − 1.01i)7-s + (0.433 + 0.900i)8-s + (1.44 + 2.63i)9-s + (0.0879 + 0.128i)10-s + (0.535 + 3.54i)11-s + (0.492 + 1.66i)12-s + (4.78 − 3.81i)13-s + (−1.90 − 1.83i)14-s + (0.114 + 0.244i)15-s + (0.0747 + 0.997i)16-s + (−1.95 − 0.602i)17-s + ⋯ |
| L(s) = 1 | + (0.658 + 0.258i)2-s + (0.860 + 0.509i)3-s + (0.366 + 0.340i)4-s + (0.0576 + 0.0393i)5-s + (0.434 + 0.557i)6-s + (−0.923 − 0.382i)7-s + (0.153 + 0.318i)8-s + (0.480 + 0.876i)9-s + (0.0278 + 0.0407i)10-s + (0.161 + 1.07i)11-s + (0.142 + 0.479i)12-s + (1.32 − 1.05i)13-s + (−0.509 − 0.490i)14-s + (0.0295 + 0.0632i)15-s + (0.0186 + 0.249i)16-s + (−0.473 − 0.146i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.619 - 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.619 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.04343 + 0.990503i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.04343 + 0.990503i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.930 - 0.365i)T \) |
| 3 | \( 1 + (-1.49 - 0.882i)T \) |
| 7 | \( 1 + (2.44 + 1.01i)T \) |
| good | 5 | \( 1 + (-0.128 - 0.0879i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (-0.535 - 3.54i)T + (-10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (-4.78 + 3.81i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (1.95 + 0.602i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (2.10 - 1.21i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.62 + 8.49i)T + (-19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (4.78 - 1.09i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-1.73 - 1.00i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.28 - 2.11i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (-9.06 + 4.36i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (4.79 + 2.30i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-2.30 + 5.86i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (0.683 - 0.736i)T + (-3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (5.97 - 4.07i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (-0.373 - 0.403i)T + (-4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (6.76 - 11.7i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.13 + 1.17i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-1.20 + 0.473i)T + (53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (-8.15 - 14.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-9.06 + 11.3i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-0.649 - 0.0978i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 + 2.73iT - 97T^{2} \) |
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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
−12.27978715556441382093366001875, −10.64052642650233507470548041229, −10.22834139963712329810520685441, −8.980925642597143597332719138858, −8.088266796254356337046727202816, −6.96799320867762735242927679599, −5.97552050342134547742825588892, −4.45804570931593092714351319668, −3.68257472332236433644811901656, −2.41798981574863932065728852506,
1.72707963307939429978387002291, 3.23430133660876734165614006049, 3.95081731678187340078964430380, 5.91956642483696715662983861141, 6.46422439173232309878006622631, 7.75233282518846643763233584864, 9.029387327666056715269099480939, 9.432282961939860730928347472922, 11.01723328595116704870935499470, 11.70870255664129164358785855316
