| L(s) = 1 | + (−1.04 − 1.20i)2-s + (0.198 + 0.127i)3-s + (−0.0771 + 0.536i)4-s + (−1.33 + 2.91i)5-s + (−0.0536 − 0.373i)6-s + (0.874 + 0.256i)7-s + (−1.95 + 1.25i)8-s + (−1.22 − 2.67i)9-s + (4.90 − 1.43i)10-s + (2.87 − 3.32i)11-s + (−0.0839 + 0.0968i)12-s + (−3.55 + 1.04i)13-s + (−0.603 − 1.32i)14-s + (−0.637 + 0.409i)15-s + (4.59 + 1.34i)16-s + (0.0287 + 0.199i)17-s + ⋯ |
| L(s) = 1 | + (−0.738 − 0.852i)2-s + (0.114 + 0.0738i)3-s + (−0.0385 + 0.268i)4-s + (−0.595 + 1.30i)5-s + (−0.0219 − 0.152i)6-s + (0.330 + 0.0970i)7-s + (−0.691 + 0.444i)8-s + (−0.407 − 0.892i)9-s + (1.54 − 0.455i)10-s + (0.868 − 1.00i)11-s + (−0.0242 + 0.0279i)12-s + (−0.986 + 0.289i)13-s + (−0.161 − 0.353i)14-s + (−0.164 + 0.105i)15-s + (1.14 + 0.337i)16-s + (0.00697 + 0.0485i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 + 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.723 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.429518 - 0.172085i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.429518 - 0.172085i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (-3.35 + 3.42i)T \) |
| good | 2 | \( 1 + (1.04 + 1.20i)T + (-0.284 + 1.97i)T^{2} \) |
| 3 | \( 1 + (-0.198 - 0.127i)T + (1.24 + 2.72i)T^{2} \) |
| 5 | \( 1 + (1.33 - 2.91i)T + (-3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (-0.874 - 0.256i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (-2.87 + 3.32i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (3.55 - 1.04i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.0287 - 0.199i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (0.498 - 3.46i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.339 - 2.36i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-3.00 + 1.93i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-2.46 - 5.39i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (1.56 - 3.42i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (4.40 + 2.83i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 8.39T + 47T^{2} \) |
| 53 | \( 1 + (-4.09 - 1.20i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-2.96 + 0.870i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (1.20 - 0.771i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-3.65 - 4.21i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (-0.868 - 1.00i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.41 + 9.84i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (12.4 - 3.65i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (0.397 + 0.869i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-9.64 - 6.20i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-1.94 + 4.25i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−18.25875757342640670881659670274, −16.95885895486291087722928177993, −14.89170883236758993968618687504, −14.50051366880621753989025483550, −11.92106928877710895985894981762, −11.25509939078102644206761019020, −9.935783050373021497878591395598, −8.495413031352829036073620882692, −6.49113510064450708961323541226, −3.17834018271689916685345129133,
4.86603750394554485763107657946, 7.27870100899071563448498861271, 8.343877245770678567496788060012, 9.492244503413592535655906678687, 11.76029222349860826868807476363, 12.90869468257890829327881388008, 14.77177260785279497043003176188, 15.93456710893742097732415366089, 17.01752601626851367753256164453, 17.52159659420164677564538283006
