| L(s) = 1 | + (−0.387 + 2.69i)2-s + (−0.141 − 0.309i)3-s + (−3.28 − 0.963i)4-s + (−0.353 + 0.306i)5-s + (0.890 − 0.261i)6-s + (4.93 − 7.67i)7-s + (−0.655 + 1.43i)8-s + (5.81 − 6.71i)9-s + (−0.689 − 1.07i)10-s + (−9.31 + 1.33i)11-s + (0.165 + 1.15i)12-s + (−15.9 + 10.2i)13-s + (18.7 + 16.2i)14-s + (0.145 + 0.0662i)15-s + (−15.1 − 9.72i)16-s + (2.33 + 7.94i)17-s + ⋯ |
| L(s) = 1 | + (−0.193 + 1.34i)2-s + (−0.0471 − 0.103i)3-s + (−0.820 − 0.240i)4-s + (−0.0707 + 0.0613i)5-s + (0.148 − 0.0435i)6-s + (0.704 − 1.09i)7-s + (−0.0819 + 0.179i)8-s + (0.646 − 0.745i)9-s + (−0.0689 − 0.107i)10-s + (−0.846 + 0.121i)11-s + (0.0138 + 0.0961i)12-s + (−1.23 + 0.790i)13-s + (1.34 + 1.16i)14-s + (0.00967 + 0.00441i)15-s + (−0.945 − 0.607i)16-s + (0.137 + 0.467i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.643101 + 0.535286i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.643101 + 0.535286i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (-22.6 - 3.86i)T \) |
| good | 2 | \( 1 + (0.387 - 2.69i)T + (-3.83 - 1.12i)T^{2} \) |
| 3 | \( 1 + (0.141 + 0.309i)T + (-5.89 + 6.80i)T^{2} \) |
| 5 | \( 1 + (0.353 - 0.306i)T + (3.55 - 24.7i)T^{2} \) |
| 7 | \( 1 + (-4.93 + 7.67i)T + (-20.3 - 44.5i)T^{2} \) |
| 11 | \( 1 + (9.31 - 1.33i)T + (116. - 34.0i)T^{2} \) |
| 13 | \( 1 + (15.9 - 10.2i)T + (70.2 - 153. i)T^{2} \) |
| 17 | \( 1 + (-2.33 - 7.94i)T + (-243. + 156. i)T^{2} \) |
| 19 | \( 1 + (-4.88 + 16.6i)T + (-303. - 195. i)T^{2} \) |
| 29 | \( 1 + (20.1 - 5.92i)T + (707. - 454. i)T^{2} \) |
| 31 | \( 1 + (-8.08 + 17.6i)T + (-629. - 726. i)T^{2} \) |
| 37 | \( 1 + (-48.8 - 42.2i)T + (194. + 1.35e3i)T^{2} \) |
| 41 | \( 1 + (12.8 + 14.8i)T + (-239. + 1.66e3i)T^{2} \) |
| 43 | \( 1 + (-13.8 + 6.30i)T + (1.21e3 - 1.39e3i)T^{2} \) |
| 47 | \( 1 + 66.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + (13.4 - 20.9i)T + (-1.16e3 - 2.55e3i)T^{2} \) |
| 59 | \( 1 + (-43.0 + 27.6i)T + (1.44e3 - 3.16e3i)T^{2} \) |
| 61 | \( 1 + (54.7 + 25.0i)T + (2.43e3 + 2.81e3i)T^{2} \) |
| 67 | \( 1 + (-70.5 - 10.1i)T + (4.30e3 + 1.26e3i)T^{2} \) |
| 71 | \( 1 + (2.48 - 17.2i)T + (-4.83e3 - 1.42e3i)T^{2} \) |
| 73 | \( 1 + (-56.1 - 16.4i)T + (4.48e3 + 2.88e3i)T^{2} \) |
| 79 | \( 1 + (-44.6 - 69.4i)T + (-2.59e3 + 5.67e3i)T^{2} \) |
| 83 | \( 1 + (92.3 + 80.0i)T + (980. + 6.81e3i)T^{2} \) |
| 89 | \( 1 + (-160. + 73.1i)T + (5.18e3 - 5.98e3i)T^{2} \) |
| 97 | \( 1 + (-71.9 + 62.3i)T + (1.33e3 - 9.31e3i)T^{2} \) |
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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
−17.46639229909268724610369514180, −16.86480643279953784428470182855, −15.39494727964788716155104475884, −14.60547812154018305956532777783, −13.20284553517118458673225145736, −11.34221806467525497762727322872, −9.551451944077321098121688561279, −7.70945396385944363386082227036, −6.91230134780693029825178938496, −4.82911154333015480853863855709,
2.46337879441383850237669113282, 5.06575209771210617266417028383, 7.930121717949516694673985465341, 9.690960749525414380415292002049, 10.80416378333172268190846677005, 12.06964067291214312890552633579, 12.99961271975182368307786449562, 14.87186684583257723639473124152, 16.12679648616573831937168596836, 17.99124623481004580761859422620
