| 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
−12.04832670915853159904262272338, −10.57487585888555946058673579951, −10.14300299082214114426920133517, −8.234188513142768325043968163713, −7.55642080938271630457716681411, −6.58890251165784720360721709045, −5.83921376627496382975941471591, −5.03435903490073979979599434808, −3.62480181286661662167705233462, −1.97250617399562239898165391448,
0.40211558770401167666432565503, 2.85650146171887747992980729378, 4.25763061829190711459664145519, 4.93536779657622848763813976831, 5.75189243146018868071686130160, 6.83793450485596394098028981889, 8.001945691648780803506528890838, 9.586053059697724824426590418817, 10.06824222660876447945785787738, 11.14155574517122673889937683672
