| L(s) = 1 | + (0.0613 − 0.268i)2-s + (2.25 − 2.82i)3-s + (7.13 + 3.43i)4-s + (−2.36 + 2.97i)5-s + (−0.620 − 0.778i)6-s + (9.47 − 15.9i)7-s + (2.73 − 3.43i)8-s + (3.10 + 13.6i)9-s + (0.653 + 0.819i)10-s + (12.3 − 54.0i)11-s + (25.7 − 12.4i)12-s + (−10.7 + 47.1i)13-s + (−3.69 − 3.52i)14-s + (3.05 + 13.3i)15-s + (38.7 + 48.6i)16-s + (−51.6 + 24.8i)17-s + ⋯ |
| L(s) = 1 | + (0.0217 − 0.0951i)2-s + (0.433 − 0.543i)3-s + (0.892 + 0.429i)4-s + (−0.211 + 0.265i)5-s + (−0.0422 − 0.0529i)6-s + (0.511 − 0.859i)7-s + (0.121 − 0.151i)8-s + (0.115 + 0.504i)9-s + (0.0206 + 0.0259i)10-s + (0.338 − 1.48i)11-s + (0.619 − 0.298i)12-s + (−0.229 + 1.00i)13-s + (−0.0706 − 0.0673i)14-s + (0.0525 + 0.230i)15-s + (0.605 + 0.759i)16-s + (−0.737 + 0.355i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.76117 - 0.328489i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.76117 - 0.328489i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-9.47 + 15.9i)T \) |
| good | 2 | \( 1 + (-0.0613 + 0.268i)T + (-7.20 - 3.47i)T^{2} \) |
| 3 | \( 1 + (-2.25 + 2.82i)T + (-6.00 - 26.3i)T^{2} \) |
| 5 | \( 1 + (2.36 - 2.97i)T + (-27.8 - 121. i)T^{2} \) |
| 11 | \( 1 + (-12.3 + 54.0i)T + (-1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (10.7 - 47.1i)T + (-1.97e3 - 953. i)T^{2} \) |
| 17 | \( 1 + (51.6 - 24.8i)T + (3.06e3 - 3.84e3i)T^{2} \) |
| 19 | \( 1 + 112.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (108. + 52.3i)T + (7.58e3 + 9.51e3i)T^{2} \) |
| 29 | \( 1 + (154. - 74.2i)T + (1.52e4 - 1.90e4i)T^{2} \) |
| 31 | \( 1 + 75.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-358. + 172. i)T + (3.15e4 - 3.96e4i)T^{2} \) |
| 41 | \( 1 + (53.9 - 67.6i)T + (-1.53e4 - 6.71e4i)T^{2} \) |
| 43 | \( 1 + (-29.5 - 37.0i)T + (-1.76e4 + 7.75e4i)T^{2} \) |
| 47 | \( 1 + (-92.7 + 406. i)T + (-9.35e4 - 4.50e4i)T^{2} \) |
| 53 | \( 1 + (-284. - 136. i)T + (9.28e4 + 1.16e5i)T^{2} \) |
| 59 | \( 1 + (429. + 539. i)T + (-4.57e4 + 2.00e5i)T^{2} \) |
| 61 | \( 1 + (221. - 106. i)T + (1.41e5 - 1.77e5i)T^{2} \) |
| 67 | \( 1 - 367.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (82.7 + 39.8i)T + (2.23e5 + 2.79e5i)T^{2} \) |
| 73 | \( 1 + (-12.3 - 53.9i)T + (-3.50e5 + 1.68e5i)T^{2} \) |
| 79 | \( 1 - 436.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-289. - 1.26e3i)T + (-5.15e5 + 2.48e5i)T^{2} \) |
| 89 | \( 1 + (-17.1 - 75.2i)T + (-6.35e5 + 3.05e5i)T^{2} \) |
| 97 | \( 1 - 1.26e3T + 9.12e5T^{2} \) |
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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
−14.87435308670627220269213418077, −13.86203939388278702772620782656, −12.85197422319411637440089260092, −11.29273813200630892559490610533, −10.80848941997263624456691542230, −8.585222749405959383404015144088, −7.54372860108541381598984349089, −6.47107745659173798076117583091, −3.91728455000848427236199299437, −2.01021694297242910710350703391,
2.28167039176107155911456164282, 4.51490741007913118341283868182, 6.19600102527756015103860602331, 7.77488153403098087503045603455, 9.235021312175848886741369442618, 10.33256976182448364493858086902, 11.74633439095913592227334712480, 12.63650631369072783375219936346, 14.70832917931625999806344038414, 15.10540654898364044805425702008
