| L(s) = 1 | + (−0.415 + 0.909i)2-s + (−0.959 − 0.281i)3-s + (−0.654 − 0.755i)4-s + (−0.841 + 0.540i)5-s + (0.654 − 0.755i)6-s + (−0.588 + 4.09i)7-s + (0.959 − 0.281i)8-s + (0.841 + 0.540i)9-s + (−0.142 − 0.989i)10-s + (1.71 + 3.74i)11-s + (0.415 + 0.909i)12-s + (−0.950 − 6.61i)13-s + (−3.47 − 2.23i)14-s + (0.959 − 0.281i)15-s + (−0.142 + 0.989i)16-s + (−1.01 + 1.17i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 + 0.643i)2-s + (−0.553 − 0.162i)3-s + (−0.327 − 0.377i)4-s + (−0.376 + 0.241i)5-s + (0.267 − 0.308i)6-s + (−0.222 + 1.54i)7-s + (0.339 − 0.0996i)8-s + (0.280 + 0.180i)9-s + (−0.0450 − 0.313i)10-s + (0.515 + 1.12i)11-s + (0.119 + 0.262i)12-s + (−0.263 − 1.83i)13-s + (−0.929 − 0.597i)14-s + (0.247 − 0.0727i)15-s + (−0.0355 + 0.247i)16-s + (−0.246 + 0.284i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0759631 - 0.323133i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0759631 - 0.323133i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.415 - 0.909i)T \) |
| 3 | \( 1 + (0.959 + 0.281i)T \) |
| 5 | \( 1 + (0.841 - 0.540i)T \) |
| 23 | \( 1 + (3.65 + 3.11i)T \) |
| good | 7 | \( 1 + (0.588 - 4.09i)T + (-6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (-1.71 - 3.74i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (0.950 + 6.61i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (1.01 - 1.17i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-2.03 - 2.35i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (2.33 - 2.69i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (6.73 - 1.97i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (9.03 + 5.80i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (8.31 - 5.34i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (3.96 + 1.16i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 - 1.01T + 47T^{2} \) |
| 53 | \( 1 + (0.927 - 6.45i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-0.344 - 2.39i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-8.74 + 2.56i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-3.02 + 6.62i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-0.258 + 0.566i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (4.71 + 5.43i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (1.65 + 11.5i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-12.6 - 8.14i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-8.17 - 2.39i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (1.31 - 0.844i)T + (40.2 - 88.2i)T^{2} \) |
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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
−10.77129704528112074155360060580, −10.07547587502546175579394075366, −9.158681199717330989927003525911, −8.271661153091137663596593696488, −7.45328813023109732247727275702, −6.53507682831247930548501778503, −5.61797781563927082245851773437, −5.03356634986783269713895156223, −3.47949818974576017647853057021, −1.97296127282872059490476826175,
0.20888135314341111612789243390, 1.59088670774034821927836314415, 3.62164527958156385872685354869, 4.05911491149442410958490800342, 5.20095418902476304246797184481, 6.70186808045908736978563633551, 7.18850996711016650938335102482, 8.417849480951127221802978179810, 9.338306188153500102507071721965, 10.01226550500012128552534023386
