| L(s) = 1 | − 25.2·2-s + (−62.6 − 108. i)3-s + 124.·4-s + (1.18e3 + 2.05e3i)5-s + (1.58e3 + 2.73e3i)6-s + (1.70e3 − 2.95e3i)7-s + 9.78e3·8-s + (1.98e3 − 3.44e3i)9-s + (−2.99e4 − 5.18e4i)10-s − 9.33e4·11-s + (−7.78e3 − 1.34e4i)12-s + (−1.29e4 + 2.23e4i)13-s + (−4.29e4 + 7.44e4i)14-s + (1.48e5 − 2.57e5i)15-s − 3.10e5·16-s + (3.12e5 − 5.40e5i)17-s + ⋯ |
| L(s) = 1 | − 1.11·2-s + (−0.446 − 0.773i)3-s + 0.242·4-s + (0.848 + 1.47i)5-s + (0.497 + 0.862i)6-s + (0.268 − 0.464i)7-s + 0.844·8-s + (0.101 − 0.175i)9-s + (−0.946 − 1.63i)10-s − 1.92·11-s + (−0.108 − 0.187i)12-s + (−0.125 + 0.217i)13-s + (−0.299 + 0.517i)14-s + (0.758 − 1.31i)15-s − 1.18·16-s + (0.906 − 1.57i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.106 - 0.994i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.106 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.425842 + 0.382755i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.425842 + 0.382755i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.68e7 - 1.48e7i)T \) |
| good | 2 | \( 1 + 25.2T + 512T^{2} \) |
| 3 | \( 1 + (62.6 + 108. i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-1.18e3 - 2.05e3i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 7 | \( 1 + (-1.70e3 + 2.95e3i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + 9.33e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + (1.29e4 - 2.23e4i)T + (-5.30e9 - 9.18e9i)T^{2} \) |
| 17 | \( 1 + (-3.12e5 + 5.40e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-1.20e5 - 2.07e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-1.04e6 - 1.80e6i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + (-9.75e5 + 1.69e6i)T + (-7.25e12 - 1.25e13i)T^{2} \) |
| 31 | \( 1 + (-1.47e5 - 2.55e5i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-1.49e6 - 2.59e6i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 + 1.70e7T + 3.27e14T^{2} \) |
| 47 | \( 1 - 2.92e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + (-4.36e7 - 7.55e7i)T + (-1.64e15 + 2.85e15i)T^{2} \) |
| 59 | \( 1 + 8.12e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + (1.93e7 - 3.35e7i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-1.31e8 - 2.28e8i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + (6.08e7 - 1.05e8i)T + (-2.29e16 - 3.97e16i)T^{2} \) |
| 73 | \( 1 + (1.08e8 - 1.88e8i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (2.06e8 - 3.58e8i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + (7.11e7 + 1.23e8i)T + (-9.34e16 + 1.61e17i)T^{2} \) |
| 89 | \( 1 + (2.01e7 + 3.48e7i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 - 1.95e8T + 7.60e17T^{2} \) |
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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
−13.97086482534281097506341935457, −13.30541192887660102232389025390, −11.46745111590808349731553623880, −10.37612128574965049941283334470, −9.680061678538985367396837324947, −7.60023314039797879478731437119, −7.14849247187738292504217448920, −5.46567722735899900564752340580, −2.71858080725541966447795692749, −1.16630931372310240942173606894,
0.35257014808331262348936066829, 1.88101953874072051029002972718, 4.81732363046316989216023269556, 5.41614782653872681808859344566, 8.021135458316072976371772618781, 8.797433491306261931926582155244, 10.10936482776151997218251571298, 10.56783159421906365575823952683, 12.60960258433583676046798296611, 13.40714905078508752569003926499
