L(s) = 1 | + (2.53 + 1.25i)2-s + (−7.07 − 4.08i)3-s + (4.85 + 6.35i)4-s + (17.0 − 9.82i)5-s + (−12.8 − 19.2i)6-s + (15.6 − 9.83i)7-s + (4.34 + 22.2i)8-s + (19.9 + 34.4i)9-s + (55.4 − 3.57i)10-s + (−29.7 − 17.1i)11-s + (−8.39 − 64.8i)12-s − 3.09i·13-s + (52.1 − 5.26i)14-s − 160.·15-s + (−16.8 + 61.7i)16-s + (−14.8 + 25.6i)17-s + ⋯ |
L(s) = 1 | + (0.896 + 0.443i)2-s + (−1.36 − 0.786i)3-s + (0.607 + 0.794i)4-s + (1.52 − 0.879i)5-s + (−0.872 − 1.30i)6-s + (0.847 − 0.531i)7-s + (0.191 + 0.981i)8-s + (0.737 + 1.27i)9-s + (1.75 − 0.113i)10-s + (−0.814 − 0.470i)11-s + (−0.202 − 1.56i)12-s − 0.0661i·13-s + (0.994 − 0.100i)14-s − 2.76·15-s + (−0.262 + 0.964i)16-s + (−0.211 + 0.365i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.88937 - 0.341468i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.88937 - 0.341468i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.53 - 1.25i)T \) |
| 7 | \( 1 + (-15.6 + 9.83i)T \) |
good | 3 | \( 1 + (7.07 + 4.08i)T + (13.5 + 23.3i)T^{2} \) |
| 5 | \( 1 + (-17.0 + 9.82i)T + (62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (29.7 + 17.1i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 3.09iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (14.8 - 25.6i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (40.0 - 23.1i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-49.7 - 86.0i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 136. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (0.290 - 0.502i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (128. - 74.0i)T + (2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 141.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 259. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (174. + 301. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (184. + 106. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (100. + 57.9i)T + (1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-345. + 199. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-893. - 515. i)T + (1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 476.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (554. - 960. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (225. + 391. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 770. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + (615. + 1.06e3i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 706.T + 9.12e5T^{2} \) |
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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
−14.33823025226148858139288701689, −13.22992477061628049643505786560, −12.86473471696076590446911418042, −11.52296595998513668940167547032, −10.47773950860523813929262868232, −8.311057036604191730516753763408, −6.78949071914508261647494819264, −5.61320224826642492280642668386, −5.00425526259837835091751741928, −1.62597726957999671798427521303,
2.31725461991407600992767223249, 4.80466372186571456559953868753, 5.61650252154704716553628556899, 6.65595958917547814699092960873, 9.650635631792153709694176667084, 10.58746922861339757268433832820, 11.15319685403056850462826234191, 12.43397460743115742269124528282, 13.71560686418139426657630261058, 14.81877921670701251769938842682