| 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
−15.10540654898364044805425702008, −14.70832917931625999806344038414, −12.63650631369072783375219936346, −11.74633439095913592227334712480, −10.33256976182448364493858086902, −9.235021312175848886741369442618, −7.77488153403098087503045603455, −6.19600102527756015103860602331, −4.51490741007913118341283868182, −2.28167039176107155911456164282,
2.01021694297242910710350703391, 3.91728455000848427236199299437, 6.47107745659173798076117583091, 7.54372860108541381598984349089, 8.585222749405959383404015144088, 10.80848941997263624456691542230, 11.29273813200630892559490610533, 12.85197422319411637440089260092, 13.86203939388278702772620782656, 14.87435308670627220269213418077
