L(s) = 1 | + (−1.41 − 2.44i)3-s + i·5-s + (1.64 + 0.952i)7-s + (−2.49 + 4.32i)9-s + (−0.926 + 0.534i)11-s + (1.40 − 3.32i)13-s + (2.44 − 1.41i)15-s + (0.318 − 0.551i)17-s + (−4.96 − 2.86i)19-s − 5.38i·21-s + (−1.90 − 3.30i)23-s − 25-s + 5.62·27-s + (−4.72 − 8.18i)29-s − 1.46i·31-s + ⋯ |
L(s) = 1 | + (−0.816 − 1.41i)3-s + 0.447i·5-s + (0.623 + 0.360i)7-s + (−0.831 + 1.44i)9-s + (−0.279 + 0.161i)11-s + (0.388 − 0.921i)13-s + (0.632 − 0.364i)15-s + (0.0772 − 0.133i)17-s + (−1.13 − 0.657i)19-s − 1.17i·21-s + (−0.398 − 0.689i)23-s − 0.200·25-s + 1.08·27-s + (−0.877 − 1.52i)29-s − 0.262i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.111i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6488200348\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6488200348\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (-1.40 + 3.32i)T \) |
good | 3 | \( 1 + (1.41 + 2.44i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (-1.64 - 0.952i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.926 - 0.534i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.318 + 0.551i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.96 + 2.86i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.90 + 3.30i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.72 + 8.18i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 1.46iT - 31T^{2} \) |
| 37 | \( 1 + (-0.655 + 0.378i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.232 - 0.133i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.318 - 0.551i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 9.44iT - 47T^{2} \) |
| 53 | \( 1 + 6.99T + 53T^{2} \) |
| 59 | \( 1 + (-0.641 - 0.370i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.09 - 3.63i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.01 + 4.04i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (8.45 + 4.88i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 3.71iT - 73T^{2} \) |
| 79 | \( 1 - 9.31T + 79T^{2} \) |
| 83 | \( 1 - 5.11iT - 83T^{2} \) |
| 89 | \( 1 + (10.8 - 6.28i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.65 + 2.11i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.571184189274704014956333157988, −8.220223649523760736025751240815, −7.904994886314761012324824847880, −6.91874934137313711451230009929, −6.15461014763035364272316090624, −5.54170899896583912972814779773, −4.39300526686235526451551561084, −2.71282136367170379247637195132, −1.81499067273240447901866152800, −0.31981793992461268694053023207,
1.65169809704248167040615075568, 3.60645866163197157502938221720, 4.24878162433682793593426295846, 5.07320049351003945936416450616, 5.76832908331126913422684790837, 6.78689771940226227223934576895, 8.038558078702292143081777672017, 8.863805499635448990670330484298, 9.587760122583437137704282420920, 10.46108825991921546403648530090