| L(s) = 1 | + (−0.313 + 2.18i)2-s + (−1.04 − 2.28i)3-s + (−2.74 − 0.804i)4-s + (0.809 + 0.934i)5-s + (5.31 − 1.56i)6-s + (−1.99 − 1.28i)7-s + (0.783 − 1.71i)8-s + (−2.17 + 2.50i)9-s + (−2.29 + 1.47i)10-s + (0.272 + 1.89i)11-s + (1.02 + 7.10i)12-s + (0.165 − 0.106i)13-s + (3.42 − 3.95i)14-s + (1.29 − 2.82i)15-s + (−1.30 − 0.840i)16-s + (1.49 − 0.439i)17-s + ⋯ |
| L(s) = 1 | + (−0.221 + 1.54i)2-s + (−0.602 − 1.31i)3-s + (−1.37 − 0.402i)4-s + (0.362 + 0.417i)5-s + (2.16 − 0.637i)6-s + (−0.754 − 0.484i)7-s + (0.277 − 0.606i)8-s + (−0.724 + 0.835i)9-s + (−0.724 + 0.465i)10-s + (0.0820 + 0.570i)11-s + (0.294 + 2.05i)12-s + (0.0458 − 0.0294i)13-s + (0.915 − 1.05i)14-s + (0.333 − 0.729i)15-s + (−0.327 − 0.210i)16-s + (0.363 − 0.106i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.583 - 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.583 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.445056 + 0.228131i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.445056 + 0.228131i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (4.66 + 1.10i)T \) |
| good | 2 | \( 1 + (0.313 - 2.18i)T + (-1.91 - 0.563i)T^{2} \) |
| 3 | \( 1 + (1.04 + 2.28i)T + (-1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (-0.809 - 0.934i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (1.99 + 1.28i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.272 - 1.89i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-0.165 + 0.106i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-1.49 + 0.439i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-7.66 - 2.24i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (4.77 - 1.40i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-0.740 + 1.62i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (2.54 - 2.93i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (0.279 + 0.322i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-1.84 - 4.05i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 2.58T + 47T^{2} \) |
| 53 | \( 1 + (-8.26 - 5.30i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (6.07 - 3.90i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-3.08 + 6.75i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (1.03 - 7.19i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.103 + 0.722i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-6.18 - 1.81i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (4.77 - 3.06i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-8.44 + 9.74i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (5.77 + 12.6i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (2.83 + 3.26i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−18.02466893188954296050417225142, −16.96960248554003786328493975397, −15.98527883257831239269722138203, −14.35278486555772366835654990338, −13.40723632514628254140624396921, −11.98260926006701731673316753474, −9.812056733683498554330663034533, −7.72554860710428263572838304629, −6.84418738425911157201013021547, −5.79843252135569723671624138854,
3.53228341736896043965276327960, 5.46982253498209470819586318730, 9.218501552410745761619759708192, 9.841604098167653297422566862426, 11.10246732328369874834716603738, 12.13075424522863219356701679534, 13.55113258071429326376048765994, 15.62455707436474189489318233907, 16.55082564510708860265170786051, 17.96557111018397409727568796709
