L(s) = 1 | + (−1 + 1.73i)4-s + (4.5 + 2.59i)7-s + (−2.5 − 2.59i)13-s + (−1.99 − 3.46i)16-s + (−3 − 1.73i)19-s + 5·25-s + (−9 + 5.19i)28-s − 8.66i·31-s + (−6 + 3.46i)37-s + (6.5 − 11.2i)43-s + (10 + 17.3i)49-s + (7 − 1.73i)52-s + (−6.5 + 11.2i)61-s + 7.99·64-s + (−10.5 + 6.06i)67-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)4-s + (1.70 + 0.981i)7-s + (−0.693 − 0.720i)13-s + (−0.499 − 0.866i)16-s + (−0.688 − 0.397i)19-s + 25-s + (−1.70 + 0.981i)28-s − 1.55i·31-s + (−0.986 + 0.569i)37-s + (0.991 − 1.71i)43-s + (1.42 + 2.47i)49-s + (0.970 − 0.240i)52-s + (−0.832 + 1.44i)61-s + 0.999·64-s + (−1.28 + 0.740i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 - 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.977235 + 0.386551i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.977235 + 0.386551i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 + (2.5 + 2.59i)T \) |
good | 2 | \( 1 + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 + (-4.5 - 2.59i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3 + 1.73i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 8.66iT - 31T^{2} \) |
| 37 | \( 1 + (6 - 3.46i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-6.5 + 11.2i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.5 - 11.2i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (10.5 - 6.06i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 1.73iT - 73T^{2} \) |
| 79 | \( 1 - 13T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (16.5 + 9.52i)T + (48.5 + 84.0i)T^{2} \) |
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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
−13.62694379516947677042840371391, −12.46427648873793078139292504378, −11.80718845771292300073660906113, −10.69394454445487923535021362642, −9.067439377155730272574142174468, −8.328349868159648652891769166710, −7.41249293760598380020186165924, −5.43648123126854160264107828925, −4.44536589169225904437065558856, −2.51377898048347983367857045794,
1.56257140765442249388862088029, 4.35334440313498098980548492336, 5.10354134532696297849656395253, 6.78510443580637797647023510823, 8.047159372711070323146310603239, 9.162552205587415278270033255433, 10.50640332215871967047634143278, 11.01560413161178701888136302047, 12.37128003121127970556062776597, 13.81279173160257393401615679087