| L(s) = 1 | + (1.22 + 0.702i)2-s + (0.988 − 0.149i)3-s + (1.01 + 1.72i)4-s + (−0.893 − 0.350i)5-s + (1.31 + 0.511i)6-s + (−0.780 + 2.52i)7-s + (0.0314 + 2.82i)8-s + (0.955 − 0.294i)9-s + (−0.850 − 1.05i)10-s + (0.222 − 0.720i)11-s + (1.25 + 1.55i)12-s + (2.77 − 0.632i)13-s + (−2.73 + 2.55i)14-s + (−0.935 − 0.213i)15-s + (−1.94 + 3.49i)16-s + (1.85 + 2.71i)17-s + ⋯ |
| L(s) = 1 | + (0.867 + 0.496i)2-s + (0.570 − 0.0860i)3-s + (0.506 + 0.862i)4-s + (−0.399 − 0.156i)5-s + (0.538 + 0.208i)6-s + (−0.295 + 0.955i)7-s + (0.0111 + 0.999i)8-s + (0.318 − 0.0982i)9-s + (−0.268 − 0.334i)10-s + (0.0669 − 0.217i)11-s + (0.363 + 0.448i)12-s + (0.769 − 0.175i)13-s + (−0.730 + 0.682i)14-s + (−0.241 − 0.0551i)15-s + (−0.487 + 0.873i)16-s + (0.449 + 0.659i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.13751 + 1.66104i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.13751 + 1.66104i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.22 - 0.702i)T \) |
| 3 | \( 1 + (-0.988 + 0.149i)T \) |
| 7 | \( 1 + (0.780 - 2.52i)T \) |
| good | 5 | \( 1 + (0.893 + 0.350i)T + (3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (-0.222 + 0.720i)T + (-9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (-2.77 + 0.632i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-1.85 - 2.71i)T + (-6.21 + 15.8i)T^{2} \) |
| 19 | \( 1 + (-3.12 - 5.40i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.49 + 2.19i)T + (-8.40 - 21.4i)T^{2} \) |
| 29 | \( 1 + (8.36 + 4.02i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (-0.789 + 1.36i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.569 + 7.59i)T + (-36.5 + 5.51i)T^{2} \) |
| 41 | \( 1 + (-0.879 + 0.701i)T + (9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (-6.53 - 5.21i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (-5.26 + 4.88i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (-0.839 + 11.2i)T + (-52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (1.11 + 2.83i)T + (-43.2 + 40.1i)T^{2} \) |
| 61 | \( 1 + (-7.76 + 0.581i)T + (60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (6.05 + 3.49i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.309 - 0.641i)T + (-44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (8.73 - 9.41i)T + (-5.45 - 72.7i)T^{2} \) |
| 79 | \( 1 + (9.99 - 5.77i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.16 + 9.49i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (0.979 + 3.17i)T + (-73.5 + 50.1i)T^{2} \) |
| 97 | \( 1 + 5.36iT - 97T^{2} \) |
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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
−11.18166501071898698704017598519, −9.900903188216437428279472688500, −8.788335561049482647115098365392, −8.149623741474885811728676230202, −7.38781956208549659517605905525, −6.04541835564500092640084040969, −5.61942536274559878198438785803, −4.06875154724347276881323265331, −3.40634614209749462029192334328, −2.10838428113376310152305314614,
1.24910845156072568286393028777, 2.93875477470675483597279363914, 3.69048862862340015234633503522, 4.59938285204607510944781306881, 5.76083462434941710974796429857, 7.14045101356795595831562756861, 7.40684065822781558850895993212, 9.053199020339940716101827103084, 9.708791455955796561035010411377, 10.75000164101472441112307026940
