| L(s) = 1 | + (0.281 + 0.959i)2-s + (1.95 − 1.69i)3-s + (−0.841 + 0.540i)4-s + (−0.542 − 2.16i)5-s + (2.18 + 1.40i)6-s + (−0.748 − 0.341i)7-s + (−0.755 − 0.654i)8-s + (0.529 − 3.68i)9-s + (1.92 − 1.13i)10-s + (4.65 + 1.36i)11-s + (−0.730 + 2.48i)12-s + (−1.40 + 0.639i)13-s + (0.117 − 0.814i)14-s + (−4.74 − 3.32i)15-s + (0.415 − 0.909i)16-s + (−2.11 + 3.28i)17-s + ⋯ |
| L(s) = 1 | + (0.199 + 0.678i)2-s + (1.13 − 0.980i)3-s + (−0.420 + 0.270i)4-s + (−0.242 − 0.970i)5-s + (0.890 + 0.572i)6-s + (−0.282 − 0.129i)7-s + (−0.267 − 0.231i)8-s + (0.176 − 1.22i)9-s + (0.609 − 0.357i)10-s + (1.40 + 0.412i)11-s + (−0.210 + 0.718i)12-s + (−0.388 + 0.177i)13-s + (0.0313 − 0.217i)14-s + (−1.22 − 0.859i)15-s + (0.103 − 0.227i)16-s + (−0.512 + 0.797i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.364i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.931 + 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.70200 - 0.321424i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.70200 - 0.321424i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.281 - 0.959i)T \) |
| 5 | \( 1 + (0.542 + 2.16i)T \) |
| 23 | \( 1 + (4.76 - 0.567i)T \) |
| good | 3 | \( 1 + (-1.95 + 1.69i)T + (0.426 - 2.96i)T^{2} \) |
| 7 | \( 1 + (0.748 + 0.341i)T + (4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (-4.65 - 1.36i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (1.40 - 0.639i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (2.11 - 3.28i)T + (-7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (-4.15 + 2.66i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-3.23 - 2.07i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (5.44 - 6.28i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-4.74 - 0.682i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.536 - 3.73i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (0.999 - 0.866i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 6.16iT - 47T^{2} \) |
| 53 | \( 1 + (5.78 + 2.64i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-0.542 - 1.18i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-8.65 + 9.98i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (4.59 + 15.6i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (9.62 - 2.82i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-3.34 - 5.20i)T + (-30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (2.95 + 6.47i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (11.8 + 1.69i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-3.58 - 4.13i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-11.2 + 1.61i)T + (93.0 - 27.3i)T^{2} \) |
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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
−12.55816877712930483627422195954, −11.65041600890076816611013764430, −9.618152384437183041529341953187, −8.960138185036438831533021109684, −8.132251774657941793467696499940, −7.22252098174310843039584520782, −6.34404924234570092839728752032, −4.69743645062703114462155879784, −3.49582183078577753797006774970, −1.58108631068604077480066121710,
2.49669603095195071277447339780, 3.51234697036857538834673685058, 4.24546921143739634323551555561, 6.00496782366285356268119163846, 7.43919280320763187298832281636, 8.685537258413761766838956108124, 9.607235155372955392191165554089, 10.09001847681153271283301298815, 11.32582284503230646325383987066, 11.96833196760558992292371513770
