| L(s) = 1 | + (1.24 + 1.43i)2-s + (−13.2 − 8.49i)3-s + (1.76 − 12.2i)4-s + (−8.32 − 3.80i)5-s + (−4.25 − 29.5i)6-s + (−1.18 + 4.02i)7-s + (45.4 − 29.1i)8-s + (68.7 + 150. i)9-s + (−4.90 − 16.7i)10-s + (−138. − 119. i)11-s + (−127. + 146. i)12-s + (162. − 47.6i)13-s + (−7.26 + 3.31i)14-s + (77.7 + 120. i)15-s + (−91.5 − 26.8i)16-s + (123. − 17.7i)17-s + ⋯ |
| L(s) = 1 | + (0.311 + 0.359i)2-s + (−1.46 − 0.943i)3-s + (0.110 − 0.765i)4-s + (−0.333 − 0.152i)5-s + (−0.118 − 0.821i)6-s + (−0.0241 + 0.0821i)7-s + (0.709 − 0.456i)8-s + (0.849 + 1.85i)9-s + (−0.0490 − 0.167i)10-s + (−1.14 − 0.990i)11-s + (−0.884 + 1.02i)12-s + (0.960 − 0.282i)13-s + (−0.0370 + 0.0169i)14-s + (0.345 + 0.537i)15-s + (−0.357 − 0.104i)16-s + (0.427 − 0.0614i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.358 + 0.933i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.469945 - 0.683534i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.469945 - 0.683534i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (167. + 501. i)T \) |
| good | 2 | \( 1 + (-1.24 - 1.43i)T + (-2.27 + 15.8i)T^{2} \) |
| 3 | \( 1 + (13.2 + 8.49i)T + (33.6 + 73.6i)T^{2} \) |
| 5 | \( 1 + (8.32 + 3.80i)T + (409. + 472. i)T^{2} \) |
| 7 | \( 1 + (1.18 - 4.02i)T + (-2.01e3 - 1.29e3i)T^{2} \) |
| 11 | \( 1 + (138. + 119. i)T + (2.08e3 + 1.44e4i)T^{2} \) |
| 13 | \( 1 + (-162. + 47.6i)T + (2.40e4 - 1.54e4i)T^{2} \) |
| 17 | \( 1 + (-123. + 17.7i)T + (8.01e4 - 2.35e4i)T^{2} \) |
| 19 | \( 1 + (-519. - 74.6i)T + (1.25e5 + 3.67e4i)T^{2} \) |
| 29 | \( 1 + (-71.1 - 494. i)T + (-6.78e5 + 1.99e5i)T^{2} \) |
| 31 | \( 1 + (676. - 434. i)T + (3.83e5 - 8.40e5i)T^{2} \) |
| 37 | \( 1 + (-1.85e3 + 845. i)T + (1.22e6 - 1.41e6i)T^{2} \) |
| 41 | \( 1 + (1.23e3 - 2.70e3i)T + (-1.85e6 - 2.13e6i)T^{2} \) |
| 43 | \( 1 + (-779. + 1.21e3i)T + (-1.42e6 - 3.10e6i)T^{2} \) |
| 47 | \( 1 + 319.T + 4.87e6T^{2} \) |
| 53 | \( 1 + (373. - 1.27e3i)T + (-6.63e6 - 4.26e6i)T^{2} \) |
| 59 | \( 1 + (-2.60e3 + 763. i)T + (1.01e7 - 6.55e6i)T^{2} \) |
| 61 | \( 1 + (454. + 707. i)T + (-5.75e6 + 1.25e7i)T^{2} \) |
| 67 | \( 1 + (-4.90e3 + 4.25e3i)T + (2.86e6 - 1.99e7i)T^{2} \) |
| 71 | \( 1 + (136. + 157. i)T + (-3.61e6 + 2.51e7i)T^{2} \) |
| 73 | \( 1 + (-99.6 + 693. i)T + (-2.72e7 - 8.00e6i)T^{2} \) |
| 79 | \( 1 + (-362. - 1.23e3i)T + (-3.27e7 + 2.10e7i)T^{2} \) |
| 83 | \( 1 + (4.45e3 - 2.03e3i)T + (3.10e7 - 3.58e7i)T^{2} \) |
| 89 | \( 1 + (-5.27e3 + 8.20e3i)T + (-2.60e7 - 5.70e7i)T^{2} \) |
| 97 | \( 1 + (-7.08e3 - 3.23e3i)T + (5.79e7 + 6.69e7i)T^{2} \) |
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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
−16.39179081259200498344592224277, −15.92247169898520923159068974764, −13.92610345141175210107023524975, −12.84051169316636859427028998167, −11.43162099129895462870202203758, −10.47758657198395688194918028702, −7.76205195535081881936182377961, −6.18540225052324701985502204602, −5.30191988783462088383841213699, −0.793269435889359755251096290871,
3.86742044843594700674797574230, 5.37521373116055115763997320391, 7.47177676865976827800683820101, 9.872773420949491448659260295025, 11.18668790881215126029267630332, 11.90018213792337444897980466127, 13.26943377596565988506499267398, 15.49816962687265310532606574038, 16.17922627735780016313449116018, 17.35119517858404241454752753549
