| L(s) = 1 | + (0.639 + 0.879i)2-s + (1.73 + 0.182i)3-s + (0.252 − 0.777i)4-s + (−1.82 − 1.28i)5-s + (0.947 + 1.64i)6-s + (0.785 + 3.69i)7-s + (2.91 − 0.946i)8-s + (0.0336 + 0.00715i)9-s + (−0.0383 − 2.43i)10-s + (−1.02 − 1.14i)11-s + (0.578 − 1.30i)12-s + (0.436 + 0.981i)13-s + (−2.74 + 3.05i)14-s + (−2.93 − 2.56i)15-s + (1.37 + 0.998i)16-s + (0.0948 + 0.0854i)17-s + ⋯ |
| L(s) = 1 | + (0.452 + 0.622i)2-s + (1.00 + 0.105i)3-s + (0.126 − 0.388i)4-s + (−0.818 − 0.574i)5-s + (0.386 + 0.669i)6-s + (0.296 + 1.39i)7-s + (1.03 − 0.334i)8-s + (0.0112 + 0.00238i)9-s + (−0.0121 − 0.768i)10-s + (−0.309 − 0.344i)11-s + (0.167 − 0.375i)12-s + (0.121 + 0.272i)13-s + (−0.734 + 0.816i)14-s + (−0.757 − 0.661i)15-s + (0.343 + 0.249i)16-s + (0.0230 + 0.0207i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.882 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.65849 + 0.415107i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.65849 + 0.415107i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.82 + 1.28i)T \) |
| 31 | \( 1 + (-3.04 - 4.66i)T \) |
| good | 2 | \( 1 + (-0.639 - 0.879i)T + (-0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (-1.73 - 0.182i)T + (2.93 + 0.623i)T^{2} \) |
| 7 | \( 1 + (-0.785 - 3.69i)T + (-6.39 + 2.84i)T^{2} \) |
| 11 | \( 1 + (1.02 + 1.14i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.436 - 0.981i)T + (-8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (-0.0948 - 0.0854i)T + (1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (6.68 + 2.97i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (6.52 - 2.11i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-7.54 + 5.48i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (-0.992 + 0.572i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.0547 - 0.521i)T + (-40.1 + 8.52i)T^{2} \) |
| 43 | \( 1 + (2.77 - 6.23i)T + (-28.7 - 31.9i)T^{2} \) |
| 47 | \( 1 + (-2.67 + 3.68i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-1.01 + 4.76i)T + (-48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-0.0850 + 0.809i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 - 7.91T + 61T^{2} \) |
| 67 | \( 1 + (-11.3 - 6.57i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-11.3 - 2.40i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (2.07 - 1.86i)T + (7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (-2.44 + 2.71i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (0.316 - 0.0332i)T + (81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (4.35 - 13.4i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-3.23 - 1.05i)T + (78.4 + 57.0i)T^{2} \) |
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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
−13.29377403729402778434194594107, −12.14310112974354153406190065062, −11.20545574144888665564585092777, −9.721941851707383241014885916428, −8.446973528320554256717649209708, −8.208120277904126090431081334426, −6.49956370204625305497610571911, −5.33958179311664448455576649917, −4.15100139031044083019198126243, −2.37203025418272815501599408962,
2.37373417251500708275651419184, 3.65565448796882382874941209978, 4.35105479592214904511913171170, 6.80678346112305025954755684292, 7.931547544819210177257975088211, 8.240392431289610264150153845165, 10.30328262574087352328324107831, 10.80570542675705848069993078052, 11.97652100033764616768699920966, 12.89493004922163543580246368050
