L(s) = 1 | + 3-s + 5-s + (0.134 − 0.232i)7-s + 9-s + (−3.05 + 5.29i)11-s + (−1.14 − 1.98i)13-s + 15-s + (−3.76 − 6.52i)17-s + (−1.88 − 3.25i)19-s + (0.134 − 0.232i)21-s + (2.57 + 4.46i)23-s + 25-s + 27-s + (−3.06 + 5.31i)29-s + (−5.25 + 9.10i)31-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + (0.0508 − 0.0880i)7-s + 0.333·9-s + (−0.922 + 1.59i)11-s + (−0.318 − 0.551i)13-s + 0.258·15-s + (−0.913 − 1.58i)17-s + (−0.431 − 0.747i)19-s + (0.0293 − 0.0508i)21-s + (0.537 + 0.930i)23-s + 0.200·25-s + 0.192·27-s + (−0.569 + 0.986i)29-s + (−0.943 + 1.63i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.552151232\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.552151232\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (-4.96 - 6.50i)T \) |
good | 7 | \( 1 + (-0.134 + 0.232i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3.05 - 5.29i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.14 + 1.98i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3.76 + 6.52i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.88 + 3.25i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.57 - 4.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.06 - 5.31i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (5.25 - 9.10i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.62 - 4.54i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.80 + 3.13i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + (6.55 - 11.3i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 + (-4.89 - 8.47i)T + (-30.5 + 52.8i)T^{2} \) |
| 71 | \( 1 + (6.02 - 10.4i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.167 - 0.289i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.90 + 8.49i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.16 - 7.20i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 1.47T + 89T^{2} \) |
| 97 | \( 1 + (2.64 + 4.58i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.002429486007286838739021117755, −7.65212367629257321966608905503, −7.31353749185264329144988692102, −6.76682957459342973903174492589, −5.41017532035786842826046063137, −4.96411205402492127990445271691, −4.21297139807059537796582464403, −2.82296766639936053283594948039, −2.52784902342162146583024003636, −1.34664004128115626744197958444,
0.38222384403601907449044159830, 2.00464830629251197438883492662, 2.45697432412665481345968948169, 3.68631652240236644835967839206, 4.20301224022801294758569924507, 5.42149852381632939747908541976, 6.01733490515086848629581628287, 6.63879604428936337908576861033, 7.82657985532152754722055442724, 8.172067728219022309598328888122