| L(s) = 1 | + (0.900 − 0.433i)2-s + (−2.07 + 1.41i)3-s + (0.623 − 0.781i)4-s + (−0.733 − 0.680i)5-s + (−1.25 + 2.17i)6-s + (−0.525 − 0.909i)7-s + (0.222 − 0.974i)8-s + (1.20 − 3.06i)9-s + (−0.955 − 0.294i)10-s + (−1.70 − 2.14i)11-s + (−0.187 + 2.50i)12-s + (−3.45 + 1.06i)13-s + (−0.867 − 0.591i)14-s + (2.47 + 0.373i)15-s + (−0.222 − 0.974i)16-s + (4.89 − 4.54i)17-s + ⋯ |
| L(s) = 1 | + (0.637 − 0.306i)2-s + (−1.19 + 0.815i)3-s + (0.311 − 0.390i)4-s + (−0.327 − 0.304i)5-s + (−0.511 + 0.886i)6-s + (−0.198 − 0.343i)7-s + (0.0786 − 0.344i)8-s + (0.400 − 1.02i)9-s + (−0.302 − 0.0932i)10-s + (−0.515 − 0.645i)11-s + (−0.0540 + 0.721i)12-s + (−0.956 + 0.295i)13-s + (−0.231 − 0.158i)14-s + (0.640 + 0.0964i)15-s + (−0.0556 − 0.243i)16-s + (1.18 − 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.489 + 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.489 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.354535 - 0.605760i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.354535 - 0.605760i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.900 + 0.433i)T \) |
| 5 | \( 1 + (0.733 + 0.680i)T \) |
| 43 | \( 1 + (-3.71 - 5.40i)T \) |
| good | 3 | \( 1 + (2.07 - 1.41i)T + (1.09 - 2.79i)T^{2} \) |
| 7 | \( 1 + (0.525 + 0.909i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.70 + 2.14i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (3.45 - 1.06i)T + (10.7 - 7.32i)T^{2} \) |
| 17 | \( 1 + (-4.89 + 4.54i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (2.58 + 6.57i)T + (-13.9 + 12.9i)T^{2} \) |
| 23 | \( 1 + (6.00 - 0.905i)T + (21.9 - 6.77i)T^{2} \) |
| 29 | \( 1 + (-3.75 - 2.55i)T + (10.5 + 26.9i)T^{2} \) |
| 31 | \( 1 + (-0.765 + 10.2i)T + (-30.6 - 4.62i)T^{2} \) |
| 37 | \( 1 + (2.03 - 3.52i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.34 - 2.09i)T + (25.5 - 32.0i)T^{2} \) |
| 47 | \( 1 + (6.32 - 7.92i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (2.77 + 0.856i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (0.712 + 3.12i)T + (-53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (-0.421 - 5.62i)T + (-60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (-3.20 - 8.16i)T + (-49.1 + 45.5i)T^{2} \) |
| 71 | \( 1 + (0.0787 + 0.0118i)T + (67.8 + 20.9i)T^{2} \) |
| 73 | \( 1 + (-7.69 + 2.37i)T + (60.3 - 41.1i)T^{2} \) |
| 79 | \( 1 + (-2.18 - 3.78i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (8.81 - 6.01i)T + (30.3 - 77.2i)T^{2} \) |
| 89 | \( 1 + (-5.33 + 3.64i)T + (32.5 - 82.8i)T^{2} \) |
| 97 | \( 1 + (8.81 + 11.0i)T + (-21.5 + 94.5i)T^{2} \) |
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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
−11.14155574517122673889937683672, −10.06824222660876447945785787738, −9.586053059697724824426590418817, −8.001945691648780803506528890838, −6.83793450485596394098028981889, −5.75189243146018868071686130160, −4.93536779657622848763813976831, −4.25763061829190711459664145519, −2.85650146171887747992980729378, −0.40211558770401167666432565503,
1.97250617399562239898165391448, 3.62480181286661662167705233462, 5.03435903490073979979599434808, 5.83921376627496382975941471591, 6.58890251165784720360721709045, 7.55642080938271630457716681411, 8.234188513142768325043968163713, 10.14300299082214114426920133517, 10.57487585888555946058673579951, 12.04832670915853159904262272338
