| L(s) = 1 | + (−1.94 + 1.24i)2-s + (−0.365 + 2.53i)3-s + (0.553 − 1.21i)4-s + (0.682 + 2.32i)5-s + (−2.46 − 5.38i)6-s + (6.72 − 5.82i)7-s + (−0.876 − 6.09i)8-s + (2.32 + 0.681i)9-s + (−4.23 − 3.66i)10-s + (−7.88 + 12.2i)11-s + (2.87 + 1.84i)12-s + (6.60 − 7.62i)13-s + (−5.79 + 19.7i)14-s + (−6.15 + 0.884i)15-s + (12.8 + 14.7i)16-s + (8.06 − 3.68i)17-s + ⋯ |
| L(s) = 1 | + (−0.971 + 0.624i)2-s + (−0.121 + 0.846i)3-s + (0.138 − 0.303i)4-s + (0.136 + 0.465i)5-s + (−0.410 − 0.898i)6-s + (0.961 − 0.832i)7-s + (−0.109 − 0.762i)8-s + (0.257 + 0.0757i)9-s + (−0.423 − 0.366i)10-s + (−0.717 + 1.11i)11-s + (0.239 + 0.154i)12-s + (0.508 − 0.586i)13-s + (−0.413 + 1.40i)14-s + (−0.410 + 0.0589i)15-s + (0.800 + 0.923i)16-s + (0.474 − 0.216i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0763 - 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0763 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.409035 + 0.441548i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.409035 + 0.441548i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (0.249 + 22.9i)T \) |
| good | 2 | \( 1 + (1.94 - 1.24i)T + (1.66 - 3.63i)T^{2} \) |
| 3 | \( 1 + (0.365 - 2.53i)T + (-8.63 - 2.53i)T^{2} \) |
| 5 | \( 1 + (-0.682 - 2.32i)T + (-21.0 + 13.5i)T^{2} \) |
| 7 | \( 1 + (-6.72 + 5.82i)T + (6.97 - 48.5i)T^{2} \) |
| 11 | \( 1 + (7.88 - 12.2i)T + (-50.2 - 110. i)T^{2} \) |
| 13 | \( 1 + (-6.60 + 7.62i)T + (-24.0 - 167. i)T^{2} \) |
| 17 | \( 1 + (-8.06 + 3.68i)T + (189. - 218. i)T^{2} \) |
| 19 | \( 1 + (25.9 + 11.8i)T + (236. + 272. i)T^{2} \) |
| 29 | \( 1 + (15.6 + 34.1i)T + (-550. + 635. i)T^{2} \) |
| 31 | \( 1 + (-4.24 - 29.5i)T + (-922. + 270. i)T^{2} \) |
| 37 | \( 1 + (7.89 - 26.8i)T + (-1.15e3 - 740. i)T^{2} \) |
| 41 | \( 1 + (-16.5 + 4.86i)T + (1.41e3 - 908. i)T^{2} \) |
| 43 | \( 1 + (38.7 + 5.56i)T + (1.77e3 + 520. i)T^{2} \) |
| 47 | \( 1 - 3.32T + 2.20e3T^{2} \) |
| 53 | \( 1 + (-0.249 + 0.216i)T + (399. - 2.78e3i)T^{2} \) |
| 59 | \( 1 + (-30.0 + 34.6i)T + (-495. - 3.44e3i)T^{2} \) |
| 61 | \( 1 + (-20.9 + 3.01i)T + (3.57e3 - 1.04e3i)T^{2} \) |
| 67 | \( 1 + (20.9 + 32.5i)T + (-1.86e3 + 4.08e3i)T^{2} \) |
| 71 | \( 1 + (83.9 - 53.9i)T + (2.09e3 - 4.58e3i)T^{2} \) |
| 73 | \( 1 + (41.3 - 90.5i)T + (-3.48e3 - 4.02e3i)T^{2} \) |
| 79 | \( 1 + (-54.3 - 47.0i)T + (888. + 6.17e3i)T^{2} \) |
| 83 | \( 1 + (-12.2 + 41.8i)T + (-5.79e3 - 3.72e3i)T^{2} \) |
| 89 | \( 1 + (77.2 + 11.1i)T + (7.60e3 + 2.23e3i)T^{2} \) |
| 97 | \( 1 + (35.2 + 120. i)T + (-7.91e3 + 5.08e3i)T^{2} \) |
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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
−17.76280416379361588255053889669, −16.90821688781702510913753655631, −15.67384108296016740152678382434, −14.77946031301525358589573471898, −12.95077501646854832463238947150, −10.68086665731319201143080556225, −10.06739127618853466815153360514, −8.294621537063119407079442145193, −7.03192607964173552047805539229, −4.49847896309041648793511314809,
1.66021864270763946643223263906, 5.65060886339555062443250142171, 8.021405880627520533969485821338, 8.975894382396026853757541869669, 10.77833930139024843180637825234, 11.87989402816015742938655416135, 13.21879407593033806048232863351, 14.76388195645442884819399250926, 16.56469289211765920666876039770, 17.84459019945620432602364394644
