| L(s) = 1 | + (1.59 − 1.83i)2-s + (−1.87 + 1.20i)3-s + (−0.272 − 1.89i)4-s + (−2.69 + 1.22i)5-s + (−0.771 + 5.36i)6-s + (−1.33 − 4.53i)7-s + (4.26 + 2.73i)8-s + (−1.67 + 3.67i)9-s + (−2.02 + 6.90i)10-s + (11.0 − 9.55i)11-s + (2.79 + 3.22i)12-s + (−22.7 − 6.68i)13-s + (−10.4 − 4.77i)14-s + (3.56 − 5.54i)15-s + (19.1 − 5.63i)16-s + (12.1 + 1.75i)17-s + ⋯ |
| L(s) = 1 | + (0.796 − 0.919i)2-s + (−0.624 + 0.401i)3-s + (−0.0682 − 0.474i)4-s + (−0.538 + 0.245i)5-s + (−0.128 + 0.894i)6-s + (−0.190 − 0.648i)7-s + (0.532 + 0.342i)8-s + (−0.186 + 0.408i)9-s + (−0.202 + 0.690i)10-s + (1.00 − 0.868i)11-s + (0.233 + 0.269i)12-s + (−1.75 − 0.513i)13-s + (−0.747 − 0.341i)14-s + (0.237 − 0.369i)15-s + (1.19 − 0.352i)16-s + (0.716 + 0.103i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 + 0.609i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.792 + 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.991784 - 0.337266i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.991784 - 0.337266i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (-5.15 - 22.4i)T \) |
| good | 2 | \( 1 + (-1.59 + 1.83i)T + (-0.569 - 3.95i)T^{2} \) |
| 3 | \( 1 + (1.87 - 1.20i)T + (3.73 - 8.18i)T^{2} \) |
| 5 | \( 1 + (2.69 - 1.22i)T + (16.3 - 18.8i)T^{2} \) |
| 7 | \( 1 + (1.33 + 4.53i)T + (-41.2 + 26.4i)T^{2} \) |
| 11 | \( 1 + (-11.0 + 9.55i)T + (17.2 - 119. i)T^{2} \) |
| 13 | \( 1 + (22.7 + 6.68i)T + (142. + 91.3i)T^{2} \) |
| 17 | \( 1 + (-12.1 - 1.75i)T + (277. + 81.4i)T^{2} \) |
| 19 | \( 1 + (-5.54 + 0.797i)T + (346. - 101. i)T^{2} \) |
| 29 | \( 1 + (3.08 - 21.4i)T + (-806. - 236. i)T^{2} \) |
| 31 | \( 1 + (0.0511 + 0.0328i)T + (399. + 874. i)T^{2} \) |
| 37 | \( 1 + (-13.3 - 6.09i)T + (896. + 1.03e3i)T^{2} \) |
| 41 | \( 1 + (-4.95 - 10.8i)T + (-1.10e3 + 1.27e3i)T^{2} \) |
| 43 | \( 1 + (27.4 + 42.6i)T + (-768. + 1.68e3i)T^{2} \) |
| 47 | \( 1 + 23.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + (28.2 + 96.2i)T + (-2.36e3 + 1.51e3i)T^{2} \) |
| 59 | \( 1 + (-34.0 - 9.99i)T + (2.92e3 + 1.88e3i)T^{2} \) |
| 61 | \( 1 + (49.9 - 77.7i)T + (-1.54e3 - 3.38e3i)T^{2} \) |
| 67 | \( 1 + (72.5 + 62.8i)T + (638. + 4.44e3i)T^{2} \) |
| 71 | \( 1 + (-35.6 + 41.1i)T + (-717. - 4.98e3i)T^{2} \) |
| 73 | \( 1 + (-6.81 - 47.3i)T + (-5.11e3 + 1.50e3i)T^{2} \) |
| 79 | \( 1 + (10.2 - 34.9i)T + (-5.25e3 - 3.37e3i)T^{2} \) |
| 83 | \( 1 + (-95.0 - 43.4i)T + (4.51e3 + 5.20e3i)T^{2} \) |
| 89 | \( 1 + (2.45 + 3.81i)T + (-3.29e3 + 7.20e3i)T^{2} \) |
| 97 | \( 1 + (-119. + 54.6i)T + (6.16e3 - 7.11e3i)T^{2} \) |
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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
−17.18822784271308461776725320586, −16.53755189726705494040137531512, −14.72252048251105479410591442193, −13.56045456150962152083653967352, −12.05037109554911537186375094591, −11.26840126642299909379336250808, −10.08083662358893388906891451374, −7.57682152792827381472437539532, −5.17654399874535466173063186251, −3.51707798217482505629092660775,
4.62795197390074098227094132564, 6.21292927699439652072709677990, 7.40468695588340325545011944009, 9.595543649412257594714026474741, 11.96481455974810033292033752014, 12.45317127327953400175966910430, 14.40282234487935757105808770181, 15.09933363105423117212647243843, 16.49992821706803998327430328622, 17.36412889734448636649126248200
