| L(s) = 1 | + (1.69 − 0.496i)3-s + (−0.630 − 0.404i)5-s + (−0.0283 − 0.197i)7-s + (0.0904 − 0.0581i)9-s + (−1.64 + 3.59i)11-s + (0.588 − 4.09i)13-s + (−1.26 − 0.371i)15-s + (1.78 + 2.06i)17-s + (−2.52 + 2.91i)19-s + (−0.145 − 0.319i)21-s + (−4.78 + 0.377i)23-s + (−1.84 − 4.03i)25-s + (−3.33 + 3.85i)27-s + (−4.13 − 4.77i)29-s + (5.69 + 1.67i)31-s + ⋯ |
| L(s) = 1 | + (0.976 − 0.286i)3-s + (−0.281 − 0.181i)5-s + (−0.0107 − 0.0745i)7-s + (0.0301 − 0.0193i)9-s + (−0.495 + 1.08i)11-s + (0.163 − 1.13i)13-s + (−0.327 − 0.0960i)15-s + (0.433 + 0.500i)17-s + (−0.580 + 0.669i)19-s + (−0.0318 − 0.0697i)21-s + (−0.996 + 0.0786i)23-s + (−0.368 − 0.807i)25-s + (−0.642 + 0.741i)27-s + (−0.767 − 0.886i)29-s + (1.02 + 0.300i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.17722 - 0.137135i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17722 - 0.137135i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (4.78 - 0.377i)T \) |
| good | 3 | \( 1 + (-1.69 + 0.496i)T + (2.52 - 1.62i)T^{2} \) |
| 5 | \( 1 + (0.630 + 0.404i)T + (2.07 + 4.54i)T^{2} \) |
| 7 | \( 1 + (0.0283 + 0.197i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (1.64 - 3.59i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.588 + 4.09i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-1.78 - 2.06i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (2.52 - 2.91i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (4.13 + 4.77i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-5.69 - 1.67i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (-6.76 + 4.34i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (-5.51 - 3.54i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-6.37 + 1.87i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 - 1.22T + 47T^{2} \) |
| 53 | \( 1 + (-0.705 - 4.90i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (-1.64 + 11.4i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (11.9 + 3.51i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (0.860 + 1.88i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (-3.58 - 7.85i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-0.122 + 0.141i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (0.139 - 0.969i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-4.66 + 2.99i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (16.2 - 4.77i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-11.6 - 7.48i)T + (40.2 + 88.2i)T^{2} \) |
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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
−14.07217524778304042491602042473, −12.97906395525829172355604996941, −12.23191943429873749397530571865, −10.63583806520366143951668148963, −9.613930989261910579431528205584, −8.098889356520693972809352303905, −7.76344728662076054487694878471, −5.86243990299274449428959368105, −4.07707823344088101802071971990, −2.39888837798450508560481424104,
2.75816976085574177475349359227, 4.09905190872417244999117899713, 5.97185065794592753111109248202, 7.56118190246535884407323955826, 8.661457577175244552670671989079, 9.478097263869950362679538548848, 10.92735637464390711730693217457, 11.87193404681880207505009273787, 13.42333768608785995664454226343, 14.08372721445037261763386541553
