L(s) = 1 | + (2 + 3.46i)5-s + (1 − 1.73i)11-s − 6·13-s + (−2 + 3.46i)17-s + (2 + 3.46i)19-s + (1 + 1.73i)23-s + (−5.49 + 9.52i)25-s + 2·29-s + (−1 − 1.73i)37-s − 4·43-s + (6 + 10.3i)47-s + (−3 + 5.19i)53-s + 7.99·55-s + (−4 + 6.92i)59-s + (−3 − 5.19i)61-s + ⋯ |
L(s) = 1 | + (0.894 + 1.54i)5-s + (0.301 − 0.522i)11-s − 1.66·13-s + (−0.485 + 0.840i)17-s + (0.458 + 0.794i)19-s + (0.208 + 0.361i)23-s + (−1.09 + 1.90i)25-s + 0.371·29-s + (−0.164 − 0.284i)37-s − 0.609·43-s + (0.875 + 1.51i)47-s + (−0.412 + 0.713i)53-s + 1.07·55-s + (−0.520 + 0.901i)59-s + (−0.384 − 0.665i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.486002552\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.486002552\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-2 - 3.46i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 6T + 13T^{2} \) |
| 17 | \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1 - 1.73i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (-6 - 10.3i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4 + 6.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 14T + 71T^{2} \) |
| 73 | \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6 + 10.3i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.696858559405354909242286375666, −8.991926932846871348188833498289, −7.72554273198192039092556032398, −7.22354982346805608354521183027, −6.27747580983298863161239921394, −5.83702378265161556127313160312, −4.68669133718757697812104937707, −3.44113408142490314830604011824, −2.69322888446809988694895972122, −1.74653402820437624488130484024,
0.52077666071310926347477479911, 1.83407674202403525348031921949, 2.72475699143067748154392115129, 4.40863314865487492874741955917, 4.92245345594503368454737758928, 5.48859671670230889333768251571, 6.70936863906001212550080772473, 7.36180216667961645816901241903, 8.474960798267070556137058409536, 9.088866418172523078742275363020