L(s) = 1 | + (1.16 + 0.804i)2-s + (0.705 + 1.87i)4-s + (1.51 + 1.64i)5-s + (3.43 − 3.43i)7-s + (−0.684 + 2.74i)8-s + (0.436 + 3.13i)10-s − 3.48·11-s + (−2.05 − 2.05i)13-s + (6.76 − 1.23i)14-s + (−3.00 + 2.64i)16-s + (1.64 + 1.64i)17-s + 0.642i·19-s + (−2.01 + 3.99i)20-s + (−4.04 − 2.80i)22-s + (−2.31 − 2.31i)23-s + ⋯ |
L(s) = 1 | + (0.822 + 0.568i)2-s + (0.352 + 0.935i)4-s + (0.676 + 0.736i)5-s + (1.29 − 1.29i)7-s + (−0.242 + 0.970i)8-s + (0.138 + 0.990i)10-s − 1.04·11-s + (−0.568 − 0.568i)13-s + (1.80 − 0.329i)14-s + (−0.751 + 0.660i)16-s + (0.400 + 0.400i)17-s + 0.147i·19-s + (−0.449 + 0.893i)20-s + (−0.863 − 0.597i)22-s + (−0.481 − 0.481i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.495 - 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.495 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.06497 + 1.19966i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.06497 + 1.19966i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.16 - 0.804i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.51 - 1.64i)T \) |
good | 7 | \( 1 + (-3.43 + 3.43i)T - 7iT^{2} \) |
| 11 | \( 1 + 3.48T + 11T^{2} \) |
| 13 | \( 1 + (2.05 + 2.05i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.64 - 1.64i)T + 17iT^{2} \) |
| 19 | \( 1 - 0.642iT - 19T^{2} \) |
| 23 | \( 1 + (2.31 + 2.31i)T + 23iT^{2} \) |
| 29 | \( 1 + 0.699T + 29T^{2} \) |
| 31 | \( 1 + 1.56iT - 31T^{2} \) |
| 37 | \( 1 + (5.31 - 5.31i)T - 37iT^{2} \) |
| 41 | \( 1 + 4.92T + 41T^{2} \) |
| 43 | \( 1 + (-3.56 + 3.56i)T - 43iT^{2} \) |
| 47 | \( 1 + (-6.85 + 6.85i)T - 47iT^{2} \) |
| 53 | \( 1 + (1.94 + 1.94i)T + 53iT^{2} \) |
| 59 | \( 1 + 2.74iT - 59T^{2} \) |
| 61 | \( 1 + 5.20iT - 61T^{2} \) |
| 67 | \( 1 + (-6.92 - 6.92i)T + 67iT^{2} \) |
| 71 | \( 1 + 11.1iT - 71T^{2} \) |
| 73 | \( 1 + (6.56 - 6.56i)T - 73iT^{2} \) |
| 79 | \( 1 - 2.09T + 79T^{2} \) |
| 83 | \( 1 + (6.64 - 6.64i)T - 83iT^{2} \) |
| 89 | \( 1 + 0.733iT - 89T^{2} \) |
| 97 | \( 1 + (-8.79 - 8.79i)T + 97iT^{2} \) |
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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
−11.59675216031657146911662421263, −10.62331256198735103372994537903, −10.18827705435938125557098359743, −8.329402998709871158731844391936, −7.63639455855124594474043111903, −6.89790472793742291846350170465, −5.60806042905997904237443187001, −4.82754607716379053690888959967, −3.56516121463976240176273575067, −2.16055233180124220718341907571,
1.73710348189737090439017303405, 2.62360482376657436819796153529, 4.56272286215160303343992039909, 5.27312894930460889006576085286, 5.86901810729897452005296343582, 7.52013915807555083102444108326, 8.712375078453709039517686273270, 9.514525443860519521995164155045, 10.53695372837989941406412554340, 11.55894069241274508430159118926