| L(s) = 1 | + (−0.997 + 0.0713i)2-s + (−0.212 + 0.977i)3-s + (0.989 − 0.142i)4-s + (1.87 − 1.22i)5-s + (0.142 − 0.989i)6-s + (−0.241 + 0.442i)7-s + (−0.977 + 0.212i)8-s + (−0.909 − 0.415i)9-s + (−1.78 + 1.35i)10-s + (−2.96 + 2.57i)11-s + (−0.0713 + 0.997i)12-s + (2.26 − 1.23i)13-s + (0.209 − 0.458i)14-s + (0.796 + 2.08i)15-s + (0.959 − 0.281i)16-s + (3.95 + 5.28i)17-s + ⋯ |
| L(s) = 1 | + (−0.705 + 0.0504i)2-s + (−0.122 + 0.564i)3-s + (0.494 − 0.0711i)4-s + (0.837 − 0.546i)5-s + (0.0580 − 0.404i)6-s + (−0.0912 + 0.167i)7-s + (−0.345 + 0.0751i)8-s + (−0.303 − 0.138i)9-s + (−0.562 + 0.427i)10-s + (−0.894 + 0.775i)11-s + (−0.0205 + 0.287i)12-s + (0.628 − 0.343i)13-s + (0.0559 − 0.122i)14-s + (0.205 + 0.539i)15-s + (0.239 − 0.0704i)16-s + (0.959 + 1.28i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 - 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.542 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.01546 + 0.553147i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01546 + 0.553147i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.997 - 0.0713i)T \) |
| 3 | \( 1 + (0.212 - 0.977i)T \) |
| 5 | \( 1 + (-1.87 + 1.22i)T \) |
| 23 | \( 1 + (-4.76 - 0.524i)T \) |
| good | 7 | \( 1 + (0.241 - 0.442i)T + (-3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (2.96 - 2.57i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.26 + 1.23i)T + (7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (-3.95 - 5.28i)T + (-4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (-0.338 - 2.35i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (10.1 + 1.45i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-8.71 - 5.60i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (2.77 + 7.44i)T + (-27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (0.135 + 0.296i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-11.1 - 2.42i)T + (39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (-1.49 - 1.49i)T + 47iT^{2} \) |
| 53 | \( 1 + (-5.54 - 3.02i)T + (28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (1.90 - 6.50i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (0.289 - 0.449i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-0.180 - 2.52i)T + (-66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (-1.17 + 1.35i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-9.28 - 6.94i)T + (20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (11.1 + 3.28i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (16.9 - 6.33i)T + (62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (8.76 - 5.63i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-4.85 - 1.81i)T + (73.3 + 63.5i)T^{2} \) |
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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
−10.40909412881124382822422627020, −9.796271771512913205069066680703, −8.990551173542970297431865240810, −8.218930238303540906731989530656, −7.26673452262593939593150508046, −5.83774299053911291909942854791, −5.53942507965587776902035105246, −4.14622833580041292501110407859, −2.72365757853523774073494790524, −1.36200131930012683906090945976,
0.879868695473455678710443434409, 2.38890399878573776896402325785, 3.25569632748757952215981006353, 5.22845831707385965342519452304, 6.00187860735921624155056619453, 6.97000641184537741871440608391, 7.61090476011488513197784200167, 8.674392395618367717460006158447, 9.490693741715886056141638614181, 10.30373259543288238208888276626
