L(s) = 1 | + (20.2 − 511. i)2-s + 1.13e4i·3-s + (−2.61e5 − 2.06e4i)4-s − 4.29e5·5-s + (5.81e6 + 2.29e5i)6-s + 2.05e7i·7-s + (−1.58e7 + 1.33e8i)8-s − 1.29e8·9-s + (−8.68e6 + 2.19e8i)10-s − 1.14e9i·11-s + (2.34e8 − 2.96e9i)12-s + 3.20e9·13-s + (1.05e10 + 4.14e8i)14-s − 4.88e9i·15-s + (6.78e10 + 1.08e10i)16-s + 1.45e11·17-s + ⋯ |
L(s) = 1 | + (0.0394 − 0.999i)2-s + 0.577i·3-s + (−0.996 − 0.0788i)4-s − 0.220·5-s + (0.576 + 0.0227i)6-s + 0.508i·7-s + (−0.118 + 0.992i)8-s − 0.333·9-s + (−0.00868 + 0.219i)10-s − 0.484i·11-s + (0.0455 − 0.575i)12-s + 0.301·13-s + (0.508 + 0.0200i)14-s − 0.127i·15-s + (0.987 + 0.157i)16-s + 1.23·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0788 + 0.996i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (-0.0788 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{19}{2})\) |
\(\approx\) |
\(1.443720315\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.443720315\) |
\(L(10)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-20.2 + 511. i)T \) |
| 3 | \( 1 - 1.13e4iT \) |
good | 5 | \( 1 + 4.29e5T + 3.81e12T^{2} \) |
| 7 | \( 1 - 2.05e7iT - 1.62e15T^{2} \) |
| 11 | \( 1 + 1.14e9iT - 5.55e18T^{2} \) |
| 13 | \( 1 - 3.20e9T + 1.12e20T^{2} \) |
| 17 | \( 1 - 1.45e11T + 1.40e22T^{2} \) |
| 19 | \( 1 + 2.89e11iT - 1.04e23T^{2} \) |
| 23 | \( 1 + 1.92e12iT - 3.24e24T^{2} \) |
| 29 | \( 1 - 5.44e12T + 2.10e26T^{2} \) |
| 31 | \( 1 + 3.60e13iT - 6.99e26T^{2} \) |
| 37 | \( 1 - 6.53e13T + 1.68e28T^{2} \) |
| 41 | \( 1 + 4.53e13T + 1.07e29T^{2} \) |
| 43 | \( 1 + 5.28e14iT - 2.52e29T^{2} \) |
| 47 | \( 1 - 1.17e15iT - 1.25e30T^{2} \) |
| 53 | \( 1 - 5.32e15T + 1.08e31T^{2} \) |
| 59 | \( 1 - 1.61e15iT - 7.50e31T^{2} \) |
| 61 | \( 1 - 1.33e16T + 1.36e32T^{2} \) |
| 67 | \( 1 + 1.51e16iT - 7.40e32T^{2} \) |
| 71 | \( 1 + 5.81e16iT - 2.10e33T^{2} \) |
| 73 | \( 1 - 9.74e16T + 3.46e33T^{2} \) |
| 79 | \( 1 - 1.13e17iT - 1.43e34T^{2} \) |
| 83 | \( 1 - 1.92e17iT - 3.49e34T^{2} \) |
| 89 | \( 1 + 6.58e17T + 1.22e35T^{2} \) |
| 97 | \( 1 - 6.80e16T + 5.77e35T^{2} \) |
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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
−15.17643093099042919303411928392, −13.77288721876211951440622715443, −12.19901656982809025565997880995, −11.02638759066576238324390554829, −9.656959740784657226450742211677, −8.335888217593122980918153873838, −5.58005638874889013045071419161, −4.00357912456899207967124184354, −2.58721733550680069997095286850, −0.59665867813059483205555438640,
1.12859807585215814678557576892, 3.73756570314956808733305713257, 5.56695404803181256834696400092, 7.10516068214223002110171636587, 8.157150751915203247362801131841, 9.949838772095582768272123495568, 12.13618526976176934031830958228, 13.51611848983617687257611874120, 14.63521758565657571597139331721, 16.10952243826861983970569928457