| L(s) = 1 | + (7.29 + 4.68i)2-s + (2.75 + 19.1i)3-s + (−21.9 − 48.1i)4-s + (−520. − 152. i)5-s + (−69.7 + 152. i)6-s + (−684. + 789. i)7-s + (223. − 1.55e3i)8-s + (1.73e3 − 510. i)9-s + (−3.07e3 − 3.55e3i)10-s + (−3.59e3 + 2.30e3i)11-s + (861. − 553. i)12-s + (−665. − 768. i)13-s + (−8.68e3 + 2.55e3i)14-s + (1.49e3 − 1.03e4i)15-s + (4.46e3 − 5.15e3i)16-s + (−4.86e3 + 1.06e4i)17-s + ⋯ |
| L(s) = 1 | + (0.644 + 0.414i)2-s + (0.0589 + 0.409i)3-s + (−0.171 − 0.375i)4-s + (−1.86 − 0.546i)5-s + (−0.131 + 0.288i)6-s + (−0.753 + 0.869i)7-s + (0.154 − 1.07i)8-s + (0.794 − 0.233i)9-s + (−0.973 − 1.12i)10-s + (−0.813 + 0.522i)11-s + (0.143 − 0.0924i)12-s + (−0.0840 − 0.0969i)13-s + (−0.846 + 0.248i)14-s + (0.114 − 0.794i)15-s + (0.272 − 0.314i)16-s + (−0.240 + 0.525i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 + 0.515i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.857 + 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0366291 - 0.132033i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0366291 - 0.132033i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (3.50e4 + 4.66e4i)T \) |
| good | 2 | \( 1 + (-7.29 - 4.68i)T + (53.1 + 116. i)T^{2} \) |
| 3 | \( 1 + (-2.75 - 19.1i)T + (-2.09e3 + 616. i)T^{2} \) |
| 5 | \( 1 + (520. + 152. i)T + (6.57e4 + 4.22e4i)T^{2} \) |
| 7 | \( 1 + (684. - 789. i)T + (-1.17e5 - 8.15e5i)T^{2} \) |
| 11 | \( 1 + (3.59e3 - 2.30e3i)T + (8.09e6 - 1.77e7i)T^{2} \) |
| 13 | \( 1 + (665. + 768. i)T + (-8.93e6 + 6.21e7i)T^{2} \) |
| 17 | \( 1 + (4.86e3 - 1.06e4i)T + (-2.68e8 - 3.10e8i)T^{2} \) |
| 19 | \( 1 + (7.16e3 + 1.56e4i)T + (-5.85e8 + 6.75e8i)T^{2} \) |
| 29 | \( 1 + (1.48e4 - 3.25e4i)T + (-1.12e10 - 1.30e10i)T^{2} \) |
| 31 | \( 1 + (-2.23e3 + 1.55e4i)T + (-2.63e10 - 7.75e9i)T^{2} \) |
| 37 | \( 1 + (3.95e5 - 1.16e5i)T + (7.98e10 - 5.13e10i)T^{2} \) |
| 41 | \( 1 + (2.12e5 + 6.24e4i)T + (1.63e11 + 1.05e11i)T^{2} \) |
| 43 | \( 1 + (1.18e5 + 8.24e5i)T + (-2.60e11 + 7.65e10i)T^{2} \) |
| 47 | \( 1 + 6.10e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + (-1.20e6 + 1.38e6i)T + (-1.67e11 - 1.16e12i)T^{2} \) |
| 59 | \( 1 + (1.16e6 + 1.34e6i)T + (-3.54e11 + 2.46e12i)T^{2} \) |
| 61 | \( 1 + (2.93e5 - 2.04e6i)T + (-3.01e12 - 8.85e11i)T^{2} \) |
| 67 | \( 1 + (-1.68e6 - 1.08e6i)T + (2.51e12 + 5.51e12i)T^{2} \) |
| 71 | \( 1 + (3.33e6 + 2.14e6i)T + (3.77e12 + 8.27e12i)T^{2} \) |
| 73 | \( 1 + (-1.21e6 - 2.65e6i)T + (-7.23e12 + 8.34e12i)T^{2} \) |
| 79 | \( 1 + (1.27e6 + 1.46e6i)T + (-2.73e12 + 1.90e13i)T^{2} \) |
| 83 | \( 1 + (4.90e6 - 1.43e6i)T + (2.28e13 - 1.46e13i)T^{2} \) |
| 89 | \( 1 + (-4.55e5 - 3.16e6i)T + (-4.24e13 + 1.24e13i)T^{2} \) |
| 97 | \( 1 + (-1.12e7 - 3.30e6i)T + (6.79e13 + 4.36e13i)T^{2} \) |
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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
−15.49150973679197377280436811556, −15.11560159217308859333142980896, −12.94314293591386238061376677110, −12.23202068953565813489802535129, −10.31112161859894747560925504735, −8.729763621902350101940661598798, −7.00065975436578669407904376417, −4.97788667860812784064825620135, −3.81331831447937448175331485278, −0.05885273020333161518417934331,
3.22567145078665682864151644121, 4.29195582391752977775169483618, 7.19713074300317321323970751027, 8.032948402473801254709827154795, 10.58600788767977669901098390291, 11.79468993202724621428804721541, 12.85231137019685676908193954554, 13.90232400889549020166631253067, 15.57474824338464116553392817491, 16.46873978243137899124132662585
