L(s) = 1 | + (1.12 − 0.857i)2-s + (0.530 − 1.92i)4-s + (−1.19 + 0.687i)5-s + (1.80 − 3.12i)7-s + (−1.05 − 2.62i)8-s + (−0.750 + 1.79i)10-s + (1.83 + 1.05i)11-s + (0.887 − 0.512i)13-s + (−0.648 − 5.06i)14-s + (−3.43 − 2.04i)16-s − 0.808·17-s + 7.43i·19-s + (0.693 + 2.66i)20-s + (2.96 − 0.380i)22-s + (1.65 + 2.86i)23-s + ⋯ |
L(s) = 1 | + (0.795 − 0.606i)2-s + (0.265 − 0.964i)4-s + (−0.532 + 0.307i)5-s + (0.682 − 1.18i)7-s + (−0.373 − 0.927i)8-s + (−0.237 + 0.567i)10-s + (0.552 + 0.319i)11-s + (0.246 − 0.142i)13-s + (−0.173 − 1.35i)14-s + (−0.859 − 0.511i)16-s − 0.196·17-s + 1.70i·19-s + (0.155 + 0.595i)20-s + (0.632 − 0.0811i)22-s + (0.345 + 0.597i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.285 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.285 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.42455 - 1.06165i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.42455 - 1.06165i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.12 + 0.857i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1.19 - 0.687i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.80 + 3.12i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.83 - 1.05i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.887 + 0.512i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 0.808T + 17T^{2} \) |
| 19 | \( 1 - 7.43iT - 19T^{2} \) |
| 23 | \( 1 + (-1.65 - 2.86i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (7.71 + 4.45i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.26 - 5.65i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 4.01iT - 37T^{2} \) |
| 41 | \( 1 + (-3.45 - 5.99i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.245 - 0.142i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.61 + 6.25i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 3.86iT - 53T^{2} \) |
| 59 | \( 1 + (-7.06 + 4.08i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.31 + 3.64i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.43 + 1.40i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 4.69T + 71T^{2} \) |
| 73 | \( 1 - 0.409T + 73T^{2} \) |
| 79 | \( 1 + (0.0456 - 0.0790i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.40 + 1.39i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 8.91T + 89T^{2} \) |
| 97 | \( 1 + (2.76 - 4.78i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.00991448812180516542173751464, −11.24248787331380219261898350960, −10.51976635848818845495437689235, −9.527698896360830396344699689204, −7.893738083810503824569408524578, −7.00228300968004708150739154667, −5.67208879534158740447939273419, −4.26564322277766182681067374269, −3.57933961320594198893280022330, −1.53582123941884095153597259643,
2.54568166440474899826445236582, 4.14247707678418446961164515530, 5.14214077537662335726137334535, 6.21365842394256770073469205381, 7.39623562112469284369376436642, 8.555038524320934623948732142743, 9.051287327275737201242262087716, 11.16025313330379978924342824918, 11.65268752824759034667571334030, 12.54752547249350260794743125981