| L(s) = 1 | + (1.23 + 0.693i)2-s + (−0.198 − 1.31i)3-s + (1.03 + 1.70i)4-s + (−3.14 − 1.23i)5-s + (0.668 − 1.76i)6-s + (−0.568 − 2.58i)7-s + (0.0934 + 2.82i)8-s + (1.17 − 0.361i)9-s + (−3.02 − 3.70i)10-s + (1.85 − 6.02i)11-s + (2.04 − 1.70i)12-s + (1.89 − 0.431i)13-s + (1.09 − 3.57i)14-s + (−1.00 + 4.38i)15-s + (−1.84 + 3.54i)16-s + (0.911 − 0.621i)17-s + ⋯ |
| L(s) = 1 | + (0.871 + 0.490i)2-s + (−0.114 − 0.760i)3-s + (0.518 + 0.854i)4-s + (−1.40 − 0.552i)5-s + (0.272 − 0.718i)6-s + (−0.214 − 0.976i)7-s + (0.0330 + 0.999i)8-s + (0.391 − 0.120i)9-s + (−0.955 − 1.17i)10-s + (0.560 − 1.81i)11-s + (0.590 − 0.492i)12-s + (0.524 − 0.119i)13-s + (0.291 − 0.956i)14-s + (−0.258 + 1.13i)15-s + (−0.461 + 0.887i)16-s + (0.221 − 0.150i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 + 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.487 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.47852 - 0.868183i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.47852 - 0.868183i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.23 - 0.693i)T \) |
| 7 | \( 1 + (0.568 + 2.58i)T \) |
| good | 3 | \( 1 + (0.198 + 1.31i)T + (-2.86 + 0.884i)T^{2} \) |
| 5 | \( 1 + (3.14 + 1.23i)T + (3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (-1.85 + 6.02i)T + (-9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (-1.89 + 0.431i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-0.911 + 0.621i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (1.56 - 0.900i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.57 - 1.75i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (1.18 - 2.45i)T + (-18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (2.65 - 4.60i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (11.4 - 0.859i)T + (36.5 - 5.51i)T^{2} \) |
| 41 | \( 1 + (-0.342 - 0.429i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (-7.19 - 5.73i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (-8.23 + 7.64i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (-3.47 - 0.260i)T + (52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (-5.41 + 2.12i)T + (43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (-5.46 + 0.409i)T + (60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (-10.8 - 6.25i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-10.4 + 5.03i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-4.67 - 4.33i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (3.05 + 5.28i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.53 - 1.26i)T + (74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (12.0 - 3.70i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 - 8.69T + 97T^{2} \) |
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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
−11.41528298959134290826069445003, −10.74831666627851336603704545631, −8.795180626708558946966684036448, −8.117535108484737227032754545288, −7.23038678666016649917829067982, −6.58348210959286521108742386008, −5.34209199285647804528655346198, −3.89407499409239524841361804842, −3.54945536070492187378143797751, −0.944935617229707611470158013973,
2.21325800470313611961622951201, 3.72658240144409755336879252690, 4.24713197591576067919338061216, 5.30303594321570324696394570159, 6.71534124280484301713600215290, 7.43321811527550523208540261558, 8.995235159998915591254344985502, 9.890336687855198764482814029451, 10.77609779992390294176532532841, 11.49705639198974574193686188042
