| 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 | \( 1 + (3.36e7 + 9.87e6i)T + (6.39e17 + 4.11e17i)T^{2} \) |
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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
−15.60346048182028750083482368782, −14.27506150284679395421638656963, −13.58777620631762955215135623232, −12.20213668253241113572177277280, −10.17704568664426440839190032031, −9.188864623114723356254284374477, −6.58038998635276554731923864136, −5.89323547546915466761192102761, −3.89625643188663448556341286573, −1.44987727703621109389015258657,
1.59992893354092155882160072568, 3.61729038463839937261258726816, 5.11819896698785609948993728234, 7.05393651723064986375292681612, 9.109917186483255972489223036121, 10.39046834495361083224175090596, 12.12051377416593815158838454926, 13.30184438460430798281195421331, 13.82141082787841215882462176149, 15.66281369332389835192107710334
