| L(s) = 1 | + (3.54 − 4.08i)2-s + (−3.02 − 21.0i)4-s + (−1.32 − 2.90i)5-s + (−29.0 + 8.52i)7-s + (−60.1 − 38.6i)8-s + (−16.5 − 4.86i)10-s + (4.08 + 4.70i)11-s + (32.2 + 9.47i)13-s + (−67.9 + 148. i)14-s + (−207. + 61.0i)16-s + (10.8 − 75.3i)17-s + (−1.73 − 12.0i)19-s + (−57.0 + 36.6i)20-s + 33.6·22-s + (−50.7 − 97.9i)23-s + ⋯ |
| L(s) = 1 | + (1.25 − 1.44i)2-s + (−0.377 − 2.62i)4-s + (−0.118 − 0.260i)5-s + (−1.56 + 0.460i)7-s + (−2.65 − 1.70i)8-s + (−0.524 − 0.153i)10-s + (0.111 + 0.129i)11-s + (0.688 + 0.202i)13-s + (−1.29 + 2.84i)14-s + (−3.24 + 0.954i)16-s + (0.154 − 1.07i)17-s + (−0.0209 − 0.145i)19-s + (−0.638 + 0.410i)20-s + 0.326·22-s + (−0.459 − 0.887i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 - 0.625i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.779 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.667172 + 1.89707i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.667172 + 1.89707i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + (50.7 + 97.9i)T \) |
| good | 2 | \( 1 + (-3.54 + 4.08i)T + (-1.13 - 7.91i)T^{2} \) |
| 5 | \( 1 + (1.32 + 2.90i)T + (-81.8 + 94.4i)T^{2} \) |
| 7 | \( 1 + (29.0 - 8.52i)T + (288. - 185. i)T^{2} \) |
| 11 | \( 1 + (-4.08 - 4.70i)T + (-189. + 1.31e3i)T^{2} \) |
| 13 | \( 1 + (-32.2 - 9.47i)T + (1.84e3 + 1.18e3i)T^{2} \) |
| 17 | \( 1 + (-10.8 + 75.3i)T + (-4.71e3 - 1.38e3i)T^{2} \) |
| 19 | \( 1 + (1.73 + 12.0i)T + (-6.58e3 + 1.93e3i)T^{2} \) |
| 29 | \( 1 + (-11.9 + 82.9i)T + (-2.34e4 - 6.87e3i)T^{2} \) |
| 31 | \( 1 + (76.3 + 49.0i)T + (1.23e4 + 2.70e4i)T^{2} \) |
| 37 | \( 1 + (8.19 - 17.9i)T + (-3.31e4 - 3.82e4i)T^{2} \) |
| 41 | \( 1 + (25.8 + 56.6i)T + (-4.51e4 + 5.20e4i)T^{2} \) |
| 43 | \( 1 + (185. - 119. i)T + (3.30e4 - 7.23e4i)T^{2} \) |
| 47 | \( 1 - 257.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (8.59 - 2.52i)T + (1.25e5 - 8.04e4i)T^{2} \) |
| 59 | \( 1 + (246. + 72.3i)T + (1.72e5 + 1.11e5i)T^{2} \) |
| 61 | \( 1 + (-656. - 422. i)T + (9.42e4 + 2.06e5i)T^{2} \) |
| 67 | \( 1 + (283. - 327. i)T + (-4.28e4 - 2.97e5i)T^{2} \) |
| 71 | \( 1 + (-66.9 + 77.2i)T + (-5.09e4 - 3.54e5i)T^{2} \) |
| 73 | \( 1 + (79.9 + 556. i)T + (-3.73e5 + 1.09e5i)T^{2} \) |
| 79 | \( 1 + (-782. - 229. i)T + (4.14e5 + 2.66e5i)T^{2} \) |
| 83 | \( 1 + (-362. + 793. i)T + (-3.74e5 - 4.32e5i)T^{2} \) |
| 89 | \( 1 + (-524. + 337. i)T + (2.92e5 - 6.41e5i)T^{2} \) |
| 97 | \( 1 + (-303. - 663. i)T + (-5.97e5 + 6.89e5i)T^{2} \) |
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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
−11.71255971926739575710110800234, −10.54944248126400327979707641601, −9.724451087118346182081952569289, −8.913959198596878635650011302278, −6.65227092486984982219742600372, −5.80885473853243690059553677903, −4.53660832563644600294131690659, −3.41855934105098556425188378677, −2.42911303891940480987223496122, −0.55624842440219609498865495315,
3.32463148129722792988138468553, 3.82966408587671809914955584227, 5.47074935927142258478716028772, 6.36344376418534571746066150657, 7.04020967508293001532019380141, 8.131379086519697167083307302139, 9.274027683808674298757191677870, 10.69314408321154292534225793407, 12.12053730867093315145649677608, 12.96457773913010367843490064244
