| L(s) = 1 | + (11.2 + 19.4i)2-s + (4 − 6.92i)4-s + (−246. + 427. i)5-s + (381.5 + 660. i)7-s + 1.16e4·8-s − 1.10e4·10-s + (−2.84e4 − 4.92e4i)11-s + (3.65e4 − 6.32e4i)13-s + (−8.56e3 + 1.48e4i)14-s + (1.28e5 + 2.23e5i)16-s − 1.68e5·17-s − 5.98e5·19-s + (1.97e3 + 3.42e3i)20-s + (6.38e5 − 1.10e6i)22-s + (1.20e6 − 2.07e6i)23-s + ⋯ |
| L(s) = 1 | + (0.496 + 0.859i)2-s + (0.00781 − 0.0135i)4-s + (−0.176 + 0.306i)5-s + (0.0600 + 0.104i)7-s + 1.00·8-s − 0.350·10-s + (−0.585 − 1.01i)11-s + (0.354 − 0.614i)13-s + (−0.0595 + 0.103i)14-s + (0.492 + 0.852i)16-s − 0.488·17-s − 1.05·19-s + (0.00276 + 0.00478i)20-s + (0.581 − 1.00i)22-s + (0.894 − 1.54i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.46407 - 0.434483i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.46407 - 0.434483i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-11.2 - 19.4i)T + (-256 + 443. i)T^{2} \) |
| 5 | \( 1 + (246. - 427. i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 7 | \( 1 + (-381.5 - 660. i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (2.84e4 + 4.92e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + (-3.65e4 + 6.32e4i)T + (-5.30e9 - 9.18e9i)T^{2} \) |
| 17 | \( 1 + 1.68e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 5.98e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + (-1.20e6 + 2.07e6i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + (2.32e6 + 4.03e6i)T + (-7.25e12 + 1.25e13i)T^{2} \) |
| 31 | \( 1 + (-9.12e5 + 1.57e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 - 1.42e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (-1.49e7 + 2.58e7i)T + (-1.63e14 - 2.83e14i)T^{2} \) |
| 43 | \( 1 + (3.87e6 + 6.71e6i)T + (-2.51e14 + 4.35e14i)T^{2} \) |
| 47 | \( 1 + (-1.54e7 - 2.68e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + 9.90e6T + 3.29e15T^{2} \) |
| 59 | \( 1 + (6.26e7 - 1.08e8i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-7.63e7 - 1.32e8i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-1.60e8 + 2.77e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + 1.09e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 6.64e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + (-5.76e7 - 9.99e7i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (3.07e8 + 5.32e8i)T + (-9.34e16 + 1.61e17i)T^{2} \) |
| 89 | \( 1 - 3.37e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + (-1.17e8 - 2.03e8i)T + (-3.80e17 + 6.58e17i)T^{2} \) |
| 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
−12.85264188810441078411855826583, −11.08298324085894133696416439044, −10.56185658764621842338796251082, −8.745738626880019683735560242230, −7.66606129347083113351472429584, −6.44576941109770765437618442481, −5.55615943962965122444971382350, −4.21002221682861906350523395200, −2.53915337452782874455703772061, −0.61110953830559006317461109041,
1.37742020021890269004177883361, 2.56226523617330613506608653449, 4.01865939153061404403754215523, 4.94483284613005454803747105552, 6.84152507584236518576777093750, 7.994990777228925649329172152978, 9.419233784689852049022165431333, 10.72169814282680978772770163201, 11.49337342841762021841551316147, 12.69342106973085515083187088764
