L(s) = 1 | + (−0.492 − 0.998i)3-s + (−3.01 + 2.64i)5-s + (2.62 − 0.300i)7-s + (1.07 − 1.39i)9-s + (−4.08 + 0.267i)11-s + (−4.36 − 4.36i)13-s + (4.12 + 1.70i)15-s + (−3.63 + 1.94i)17-s + (0.826 + 6.28i)19-s + (−1.59 − 2.47i)21-s + (−6.53 − 3.22i)23-s + (1.44 − 10.9i)25-s + (−5.19 − 1.03i)27-s + (−0.361 − 1.81i)29-s + (−2.27 + 1.12i)31-s + ⋯ |
L(s) = 1 | + (−0.284 − 0.576i)3-s + (−1.34 + 1.18i)5-s + (0.993 − 0.113i)7-s + (0.357 − 0.465i)9-s + (−1.23 + 0.0806i)11-s + (−1.21 − 1.21i)13-s + (1.06 + 0.441i)15-s + (−0.881 + 0.472i)17-s + (0.189 + 1.44i)19-s + (−0.348 − 0.540i)21-s + (−1.36 − 0.672i)23-s + (0.289 − 2.19i)25-s + (−1.00 − 0.199i)27-s + (−0.0672 − 0.337i)29-s + (−0.408 + 0.201i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0312i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0312i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.000854658 + 0.0546865i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.000854658 + 0.0546865i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.62 + 0.300i)T \) |
| 17 | \( 1 + (3.63 - 1.94i)T \) |
good | 3 | \( 1 + (0.492 + 0.998i)T + (-1.82 + 2.38i)T^{2} \) |
| 5 | \( 1 + (3.01 - 2.64i)T + (0.652 - 4.95i)T^{2} \) |
| 11 | \( 1 + (4.08 - 0.267i)T + (10.9 - 1.43i)T^{2} \) |
| 13 | \( 1 + (4.36 + 4.36i)T + 13iT^{2} \) |
| 19 | \( 1 + (-0.826 - 6.28i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (6.53 + 3.22i)T + (14.0 + 18.2i)T^{2} \) |
| 29 | \( 1 + (0.361 + 1.81i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (2.27 - 1.12i)T + (18.8 - 24.5i)T^{2} \) |
| 37 | \( 1 + (0.379 - 5.78i)T + (-36.6 - 4.82i)T^{2} \) |
| 41 | \( 1 + (0.175 - 0.882i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (0.668 + 1.61i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (0.151 - 0.563i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (1.96 + 2.56i)T + (-13.7 + 51.1i)T^{2} \) |
| 59 | \( 1 + (4.58 + 0.603i)T + (56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (0.828 + 2.43i)T + (-48.3 + 37.1i)T^{2} \) |
| 67 | \( 1 + (-7.57 - 4.37i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.54 + 2.36i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (2.01 + 0.684i)T + (57.9 + 44.4i)T^{2} \) |
| 79 | \( 1 + (-0.497 + 1.00i)T + (-48.0 - 62.6i)T^{2} \) |
| 83 | \( 1 + (-1.61 + 3.89i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-15.9 - 4.28i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (3.12 - 0.622i)T + (89.6 - 37.1i)T^{2} \) |
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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
−10.51729044310615437733166924797, −10.15877764512810182608374275216, −8.065505451384246394683210611593, −7.911667458860432727696672250040, −7.10709811535333560072027438895, −6.00114378315937230171653118641, −4.68583822560729407546693416093, −3.57877880097935147574404241901, −2.26059845110375528829256664357, −0.03253669509882892769595033613,
2.14312392177606298945802490208, 4.14525130707566292568301878842, 4.77697469780394291256135162390, 5.22468249560124210103266955870, 7.33039004460274375941629810089, 7.73118733184223412296087559227, 8.790380971348755577335378681181, 9.536157691564584294643803923850, 10.83922008710217198727688481687, 11.43411449831922130592842625413