L(s) = 1 | + (−1 − 1.73i)2-s + (−1.99 + 3.46i)4-s + (5.04 + 8.74i)5-s + (−5.48 + 17.6i)7-s + 7.99·8-s + (10.0 − 17.4i)10-s + (−18.0 + 31.3i)11-s − 4.01·13-s + (36.1 − 8.18i)14-s + (−8 − 13.8i)16-s + (−18.9 + 32.8i)17-s + (−28.5 − 49.3i)19-s − 40.3·20-s + 72.2·22-s + (−39.2 − 67.9i)23-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.451 + 0.782i)5-s + (−0.296 + 0.955i)7-s + 0.353·8-s + (0.319 − 0.553i)10-s + (−0.495 + 0.858i)11-s − 0.0857·13-s + (0.689 − 0.156i)14-s + (−0.125 − 0.216i)16-s + (−0.270 + 0.469i)17-s + (−0.344 − 0.596i)19-s − 0.451·20-s + 0.700·22-s + (−0.355 − 0.615i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.356i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.934 - 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.4098036017\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4098036017\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + 1.73i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (5.48 - 17.6i)T \) |
good | 5 | \( 1 + (-5.04 - 8.74i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (18.0 - 31.3i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 4.01T + 2.19e3T^{2} \) |
| 17 | \( 1 + (18.9 - 32.8i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (28.5 + 49.3i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (39.2 + 67.9i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 215.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-91.1 + 157. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (156. + 270. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 186.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 262.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-68.9 - 119. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (273. - 474. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (1.86 - 3.23i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-57.5 - 99.5i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (313. - 543. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 533.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-149. + 258. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-433. - 750. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 244.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-223. - 387. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.80e3T + 9.12e5T^{2} \) |
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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
−11.15035222239618739687772808270, −10.44887611426377016650328289353, −9.605288629572818471722349590266, −8.836792501903453522543595386826, −7.68546680948967103098469643158, −6.62673391107403860653780994655, −5.59706134154193932955221772414, −4.22412896915155233071957611361, −2.72092022479224643587743597031, −2.07583499518602712202093137218,
0.15306492401934372753366707601, 1.48871774508796350006167096584, 3.45934114131511748812381403713, 4.81556986809424377060761960334, 5.70921514209345825440232504940, 6.76640694127518649088407973407, 7.77255463549714144652268250563, 8.634494161472169664534350519034, 9.539562525996404303128706078890, 10.33499393866870573054280460649