L(s) = 1 | + (15.2 − 9.83i)2-s + (−1.23 + 8.60i)3-s + (−75.3 + 164. i)4-s + (1.46e3 − 430. i)5-s + (65.6 + 143. i)6-s + (−3.09e3 − 3.57e3i)7-s + (1.79e3 + 1.24e4i)8-s + (1.88e4 + 5.52e3i)9-s + (1.81e4 − 2.09e4i)10-s + (5.47e4 + 3.52e4i)11-s + (−1.32e3 − 852. i)12-s + (6.58e4 − 7.60e4i)13-s + (−8.24e4 − 2.42e4i)14-s + (1.88e3 + 1.31e4i)15-s + (8.93e4 + 1.03e5i)16-s + (−2.12e3 − 4.65e3i)17-s + ⋯ |
L(s) = 1 | + (0.676 − 0.434i)2-s + (−0.00881 + 0.0613i)3-s + (−0.147 + 0.322i)4-s + (1.04 − 0.307i)5-s + (0.0206 + 0.0452i)6-s + (−0.487 − 0.562i)7-s + (0.154 + 1.07i)8-s + (0.955 + 0.280i)9-s + (0.575 − 0.663i)10-s + (1.12 + 0.725i)11-s + (−0.0184 − 0.0118i)12-s + (0.639 − 0.738i)13-s + (−0.573 − 0.168i)14-s + (0.00963 + 0.0670i)15-s + (0.340 + 0.393i)16-s + (−0.00617 − 0.0135i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.157i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(2.96204 - 0.233994i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.96204 - 0.233994i\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 + (3.44e5 - 1.29e6i)T \) |
good | 2 | \( 1 + (-15.2 + 9.83i)T + (212. - 465. i)T^{2} \) |
| 3 | \( 1 + (1.23 - 8.60i)T + (-1.88e4 - 5.54e3i)T^{2} \) |
| 5 | \( 1 + (-1.46e3 + 430. i)T + (1.64e6 - 1.05e6i)T^{2} \) |
| 7 | \( 1 + (3.09e3 + 3.57e3i)T + (-5.74e6 + 3.99e7i)T^{2} \) |
| 11 | \( 1 + (-5.47e4 - 3.52e4i)T + (9.79e8 + 2.14e9i)T^{2} \) |
| 13 | \( 1 + (-6.58e4 + 7.60e4i)T + (-1.50e9 - 1.04e10i)T^{2} \) |
| 17 | \( 1 + (2.12e3 + 4.65e3i)T + (-7.76e10 + 8.96e10i)T^{2} \) |
| 19 | \( 1 + (-3.42e5 + 7.49e5i)T + (-2.11e11 - 2.43e11i)T^{2} \) |
| 29 | \( 1 + (4.76e5 + 1.04e6i)T + (-9.50e12 + 1.09e13i)T^{2} \) |
| 31 | \( 1 + (-1.23e6 - 8.59e6i)T + (-2.53e13 + 7.44e12i)T^{2} \) |
| 37 | \( 1 + (2.06e7 + 6.05e6i)T + (1.09e14 + 7.02e13i)T^{2} \) |
| 41 | \( 1 + (2.52e7 - 7.42e6i)T + (2.75e14 - 1.76e14i)T^{2} \) |
| 43 | \( 1 + (-2.67e6 + 1.85e7i)T + (-4.82e14 - 1.41e14i)T^{2} \) |
| 47 | \( 1 - 5.25e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + (1.99e7 + 2.30e7i)T + (-4.69e14 + 3.26e15i)T^{2} \) |
| 59 | \( 1 + (2.62e7 - 3.03e7i)T + (-1.23e15 - 8.57e15i)T^{2} \) |
| 61 | \( 1 + (1.00e7 + 7.00e7i)T + (-1.12e16 + 3.29e15i)T^{2} \) |
| 67 | \( 1 + (2.50e7 - 1.60e7i)T + (1.13e16 - 2.47e16i)T^{2} \) |
| 71 | \( 1 + (1.76e8 - 1.13e8i)T + (1.90e16 - 4.17e16i)T^{2} \) |
| 73 | \( 1 + (-8.25e7 + 1.80e8i)T + (-3.85e16 - 4.44e16i)T^{2} \) |
| 79 | \( 1 + (-3.00e8 + 3.47e8i)T + (-1.70e16 - 1.18e17i)T^{2} \) |
| 83 | \( 1 + (2.34e8 + 6.87e7i)T + (1.57e17 + 1.01e17i)T^{2} \) |
| 89 | \( 1 + (4.08e7 - 2.83e8i)T + (-3.36e17 - 9.87e16i)T^{2} \) |
| 97 | \( 1 + (3.36e7 - 9.87e6i)T + (6.39e17 - 4.11e17i)T^{2} \) |
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
−15.66281369332389835192107710334, −13.82141082787841215882462176149, −13.30184438460430798281195421331, −12.12051377416593815158838454926, −10.39046834495361083224175090596, −9.109917186483255972489223036121, −7.05393651723064986375292681612, −5.11819896698785609948993728234, −3.61729038463839937261258726816, −1.59992893354092155882160072568,
1.44987727703621109389015258657, 3.89625643188663448556341286573, 5.89323547546915466761192102761, 6.58038998635276554731923864136, 9.188864623114723356254284374477, 10.17704568664426440839190032031, 12.20213668253241113572177277280, 13.58777620631762955215135623232, 14.27506150284679395421638656963, 15.60346048182028750083482368782