L(s) = 1 | − 3-s − 5-s + (0.278 − 0.482i)7-s + 9-s + (−0.747 + 1.29i)11-s + (−1.66 − 2.88i)13-s + 15-s + (−3.16 − 5.48i)17-s + (−2.84 − 4.92i)19-s + (−0.278 + 0.482i)21-s + (2.62 + 4.55i)23-s + 25-s − 27-s + (−1.03 + 1.79i)29-s + (−3.14 + 5.45i)31-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + (0.105 − 0.182i)7-s + 0.333·9-s + (−0.225 + 0.390i)11-s + (−0.462 − 0.800i)13-s + 0.258·15-s + (−0.768 − 1.33i)17-s + (−0.652 − 1.13i)19-s + (−0.0607 + 0.105i)21-s + (0.548 + 0.949i)23-s + 0.200·25-s − 0.192·27-s + (−0.192 + 0.332i)29-s + (−0.565 + 0.979i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.153 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.153 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4852880395\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4852880395\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-4.44 - 6.87i)T \) |
good | 7 | \( 1 + (-0.278 + 0.482i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.747 - 1.29i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.66 + 2.88i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3.16 + 5.48i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.84 + 4.92i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.62 - 4.55i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.03 - 1.79i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.14 - 5.45i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.16 - 7.21i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.52 + 6.10i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 + (-3.75 + 6.50i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 0.994T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 + (1.32 + 2.30i)T + (-30.5 + 52.8i)T^{2} \) |
| 71 | \( 1 + (-1.28 + 2.22i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (6.29 + 10.9i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.62 - 8.01i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.13 + 1.96i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 4.78T + 89T^{2} \) |
| 97 | \( 1 + (-3.17 - 5.49i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.691184008970829934716957800808, −7.75040998528820329480177955530, −7.04079231449122429776062473119, −6.74891042702656900627730507045, −5.38086531344792647817429298291, −5.04629912882070962115911720934, −4.24784418934989583067037454382, −3.17902478041122660983595477831, −2.31231631292664353741471818987, −0.892949516534672800057227962585,
0.18781854688580996997563184375, 1.67410510812476550791816835990, 2.56834428924837104585380538354, 4.01524684289885735797414898784, 4.21193674292178583781878535330, 5.33797937073447734390520020379, 6.08161539546402679620838765728, 6.65498310372778321081853106833, 7.50894805427295430191483247469, 8.303878167013378803882257136953