L(s) = 1 | − 980. i·2-s − 4.20e4i·3-s − 4.37e5·4-s + (−2.12e6 − 3.81e6i)5-s − 4.12e7·6-s − 4.16e7i·7-s − 8.52e7i·8-s − 6.07e8·9-s + (−3.74e9 + 2.08e9i)10-s + 1.14e10·11-s + 1.83e10i·12-s + 4.17e10i·13-s − 4.08e10·14-s + (−1.60e11 + 8.93e10i)15-s − 3.12e11·16-s − 7.42e9i·17-s + ⋯ |
L(s) = 1 | − 1.35i·2-s − 1.23i·3-s − 0.834·4-s + (−0.486 − 0.873i)5-s − 1.67·6-s − 0.389i·7-s − 0.224i·8-s − 0.522·9-s + (−1.18 + 0.658i)10-s + 1.46·11-s + 1.02i·12-s + 1.09i·13-s − 0.528·14-s + (−1.07 + 0.600i)15-s − 1.13·16-s − 0.0151i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.486 - 0.873i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(10)\) |
\(\approx\) |
\(0.811937 + 1.38103i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.811937 + 1.38103i\) |
\(L(\frac{21}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (2.12e6 + 3.81e6i)T \) |
good | 2 | \( 1 + 980. iT - 5.24e5T^{2} \) |
| 3 | \( 1 + 4.20e4iT - 1.16e9T^{2} \) |
| 7 | \( 1 + 4.16e7iT - 1.13e16T^{2} \) |
| 11 | \( 1 - 1.14e10T + 6.11e19T^{2} \) |
| 13 | \( 1 - 4.17e10iT - 1.46e21T^{2} \) |
| 17 | \( 1 + 7.42e9iT - 2.39e23T^{2} \) |
| 19 | \( 1 - 2.13e12T + 1.97e24T^{2} \) |
| 23 | \( 1 + 9.51e12iT - 7.46e25T^{2} \) |
| 29 | \( 1 + 5.19e13T + 6.10e27T^{2} \) |
| 31 | \( 1 + 2.16e14T + 2.16e28T^{2} \) |
| 37 | \( 1 - 7.79e14iT - 6.24e29T^{2} \) |
| 41 | \( 1 - 1.01e15T + 4.39e30T^{2} \) |
| 43 | \( 1 + 3.59e15iT - 1.08e31T^{2} \) |
| 47 | \( 1 + 5.35e14iT - 5.88e31T^{2} \) |
| 53 | \( 1 - 1.44e16iT - 5.77e32T^{2} \) |
| 59 | \( 1 - 1.09e17T + 4.42e33T^{2} \) |
| 61 | \( 1 + 1.12e17T + 8.34e33T^{2} \) |
| 67 | \( 1 - 5.23e16iT - 4.95e34T^{2} \) |
| 71 | \( 1 - 2.62e17T + 1.49e35T^{2} \) |
| 73 | \( 1 + 4.11e17iT - 2.53e35T^{2} \) |
| 79 | \( 1 - 6.87e17T + 1.13e36T^{2} \) |
| 83 | \( 1 - 1.74e18iT - 2.90e36T^{2} \) |
| 89 | \( 1 + 4.41e17T + 1.09e37T^{2} \) |
| 97 | \( 1 + 8.34e18iT - 5.60e37T^{2} \) |
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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
−18.52214884704932688327579981167, −16.63921174625562846936507669615, −13.76398795118287928331622951895, −12.38373518719507068119067389040, −11.58761331368644724761862556041, −9.204449692492887514742485098509, −7.04473472253404703331352963041, −3.98130355863727855174632213388, −1.69129196686179634757714001565, −0.78821774480713855965924585185,
3.62906517344969903661762936904, 5.57642900716856323219850371192, 7.39573164421913238834310301911, 9.365915205063801798425055196222, 11.29607486473740556204045362369, 14.45043091182218774186611200934, 15.30389035058108066650282889908, 16.30026309187555320366628805548, 17.89186585928418814256013491476, 20.01982005656872311992187391774