| L(s) = 1 | + (−0.188 − 0.165i)2-s + (1.73 − 0.0193i)3-s + (−0.244 − 1.92i)4-s + (2.54 + 0.370i)5-s + (−0.329 − 0.283i)6-s + (−0.452 − 4.99i)7-s + (−0.554 + 0.817i)8-s + (2.99 − 0.0671i)9-s + (−0.418 − 0.492i)10-s + (−3.23 + 3.29i)11-s + (−0.460 − 3.32i)12-s + (4.91 + 0.177i)13-s + (−0.743 + 1.01i)14-s + (4.42 + 0.592i)15-s + (−3.51 + 0.906i)16-s + (−2.48 + 1.15i)17-s + ⋯ |
| L(s) = 1 | + (−0.133 − 0.117i)2-s + (0.999 − 0.0111i)3-s + (−0.122 − 0.960i)4-s + (1.13 + 0.165i)5-s + (−0.134 − 0.115i)6-s + (−0.171 − 1.88i)7-s + (−0.196 + 0.289i)8-s + (0.999 − 0.0223i)9-s + (−0.132 − 0.155i)10-s + (−0.975 + 0.993i)11-s + (−0.132 − 0.959i)12-s + (1.36 + 0.0492i)13-s + (−0.198 + 0.271i)14-s + (1.14 + 0.153i)15-s + (−0.877 + 0.226i)16-s + (−0.603 + 0.279i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.346 + 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.346 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.70685 - 1.18890i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.70685 - 1.18890i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.73 + 0.0193i)T \) |
| 59 | \( 1 + (-6.75 + 3.64i)T \) |
| good | 2 | \( 1 + (0.188 + 0.165i)T + (0.252 + 1.98i)T^{2} \) |
| 5 | \( 1 + (-2.54 - 0.370i)T + (4.79 + 1.42i)T^{2} \) |
| 7 | \( 1 + (0.452 + 4.99i)T + (-6.88 + 1.25i)T^{2} \) |
| 11 | \( 1 + (3.23 - 3.29i)T + (-0.198 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-4.91 - 0.177i)T + (12.9 + 0.938i)T^{2} \) |
| 17 | \( 1 + (2.48 - 1.15i)T + (11.0 - 12.9i)T^{2} \) |
| 19 | \( 1 + (3.89 + 0.858i)T + (17.2 + 7.97i)T^{2} \) |
| 23 | \( 1 + (2.62 + 0.990i)T + (17.2 + 15.2i)T^{2} \) |
| 29 | \( 1 + (-1.86 - 9.28i)T + (-26.7 + 11.2i)T^{2} \) |
| 31 | \( 1 + (-0.790 + 2.49i)T + (-25.3 - 17.8i)T^{2} \) |
| 37 | \( 1 + (-6.67 - 9.84i)T + (-13.6 + 34.3i)T^{2} \) |
| 41 | \( 1 + (2.64 + 2.16i)T + (8.08 + 40.1i)T^{2} \) |
| 43 | \( 1 + (-7.65 + 1.97i)T + (37.6 - 20.8i)T^{2} \) |
| 47 | \( 1 + (-3.13 + 0.455i)T + (45.0 - 13.3i)T^{2} \) |
| 53 | \( 1 + (-4.95 + 5.83i)T + (-8.57 - 52.3i)T^{2} \) |
| 61 | \( 1 + (2.43 + 2.14i)T + (7.68 + 60.5i)T^{2} \) |
| 67 | \( 1 + (2.60 + 5.36i)T + (-41.5 + 52.5i)T^{2} \) |
| 71 | \( 1 + (1.21 - 3.04i)T + (-51.5 - 48.8i)T^{2} \) |
| 73 | \( 1 + (-1.24 - 0.135i)T + (71.2 + 15.6i)T^{2} \) |
| 79 | \( 1 + (0.0118 - 0.656i)T + (-78.9 - 2.85i)T^{2} \) |
| 83 | \( 1 + (1.63 - 0.829i)T + (49.0 - 66.9i)T^{2} \) |
| 89 | \( 1 + (0.614 + 0.207i)T + (70.8 + 53.8i)T^{2} \) |
| 97 | \( 1 + (-7.31 - 9.99i)T + (-29.3 + 92.4i)T^{2} \) |
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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
−10.34692946165678586861212426904, −10.05607115028248197189887174505, −9.087601006341424723521091872715, −8.093453354646321681440193944419, −6.92214173029008359069646974733, −6.31931340012517675707446766194, −4.83402795486460770429367323924, −3.92993453286901481685546143723, −2.32812847921543881037301783531, −1.30023983707101248258517220188,
2.27751398393557798969191945151, 2.76566745205209552366521592470, 4.11906590737423679191811626574, 5.72696646046160468400535964752, 6.22723569025709989578638428004, 7.83480492136575238200032362148, 8.609022754927317785009732676835, 8.908692820752104462363073215634, 9.744760736026966134401727981441, 10.97410514972850641449011026410
