| L(s) = 1 | + (−1.84 − 0.118i)2-s + (−3.90 + 5.60i)3-s + (−4.53 − 0.584i)4-s + (−3.25 + 2.12i)5-s + (7.88 − 9.89i)6-s + (29.1 − 1.87i)7-s + (22.8 + 4.45i)8-s + (−6.79 − 18.4i)9-s + (6.27 − 3.53i)10-s + (5.84 − 19.6i)11-s + (20.9 − 23.1i)12-s + (0.543 − 16.9i)13-s − 54.1·14-s + (0.851 − 26.5i)15-s + (−6.34 − 1.66i)16-s + (16.8 + 16.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.653 − 0.0419i)2-s + (−0.752 + 1.07i)3-s + (−0.566 − 0.0730i)4-s + (−0.291 + 0.189i)5-s + (0.536 − 0.673i)6-s + (1.57 − 0.101i)7-s + (1.00 + 0.196i)8-s + (−0.251 − 0.683i)9-s + (0.198 − 0.111i)10-s + (0.160 − 0.539i)11-s + (0.504 − 0.555i)12-s + (0.0115 − 0.361i)13-s − 1.03·14-s + (0.0146 − 0.456i)15-s + (−0.0991 − 0.0260i)16-s + (0.240 + 0.233i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.208 - 0.977i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.208 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.523773 + 0.647350i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.523773 + 0.647350i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 + (2.62e3 + 857. i)T \) |
| good | 2 | \( 1 + (1.84 + 0.118i)T + (7.93 + 1.02i)T^{2} \) |
| 3 | \( 1 + (3.90 - 5.60i)T + (-9.32 - 25.3i)T^{2} \) |
| 5 | \( 1 + (3.25 - 2.12i)T + (50.5 - 114. i)T^{2} \) |
| 7 | \( 1 + (-29.1 + 1.87i)T + (340. - 43.8i)T^{2} \) |
| 11 | \( 1 + (-5.84 + 19.6i)T + (-1.11e3 - 726. i)T^{2} \) |
| 13 | \( 1 + (-0.543 + 16.9i)T + (-2.19e3 - 140. i)T^{2} \) |
| 17 | \( 1 + (-16.8 - 16.3i)T + (157. + 4.91e3i)T^{2} \) |
| 19 | \( 1 + (-78.1 - 37.6i)T + (4.27e3 + 5.36e3i)T^{2} \) |
| 23 | \( 1 + (-7.88 - 48.7i)T + (-1.15e4 + 3.83e3i)T^{2} \) |
| 29 | \( 1 + (43.7 - 48.2i)T + (-2.34e3 - 2.42e4i)T^{2} \) |
| 31 | \( 1 + (-222. + 90.1i)T + (2.14e4 - 2.07e4i)T^{2} \) |
| 37 | \( 1 + (102. - 26.9i)T + (4.41e4 - 2.48e4i)T^{2} \) |
| 41 | \( 1 + (-112. - 109. i)T + (2.20e3 + 6.88e4i)T^{2} \) |
| 43 | \( 1 + (87.6 - 295. i)T + (-6.66e4 - 4.33e4i)T^{2} \) |
| 47 | \( 1 + (126. + 243. i)T + (-5.93e4 + 8.51e4i)T^{2} \) |
| 53 | \( 1 + (-16.0 - 166. i)T + (-1.46e5 + 2.84e4i)T^{2} \) |
| 59 | \( 1 + (-18.2 - 10.2i)T + (1.06e5 + 1.75e5i)T^{2} \) |
| 61 | \( 1 + (-157. - 225. i)T + (-7.83e4 + 2.13e5i)T^{2} \) |
| 67 | \( 1 + (-132. - 253. i)T + (-1.72e5 + 2.46e5i)T^{2} \) |
| 71 | \( 1 + (-250. - 680. i)T + (-2.72e5 + 2.32e5i)T^{2} \) |
| 73 | \( 1 + (-580. + 152. i)T + (3.38e5 - 1.90e5i)T^{2} \) |
| 79 | \( 1 + (629. + 409. i)T + (1.99e5 + 4.50e5i)T^{2} \) |
| 83 | \( 1 + (181. - 87.6i)T + (3.56e5 - 4.47e5i)T^{2} \) |
| 89 | \( 1 + (51.8 + 20.9i)T + (5.06e5 + 4.90e5i)T^{2} \) |
| 97 | \( 1 + (152. - 251. i)T + (-4.22e5 - 8.09e5i)T^{2} \) |
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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
−11.69919663331466956299424335989, −11.19609456410636164659018466379, −10.34833437513532974635568674322, −9.539433428870727892337606354943, −8.326826234322660830510646538920, −7.63511034122059763048917691650, −5.59073334406214201669389453786, −4.87238052760350544500372840681, −3.81810786221146059918818431904, −1.19533846375108161082384430358,
0.64737227994097425712550961505, 1.74391961825812210884572232479, 4.40740262851093328390138295849, 5.30831537430726747893056894794, 6.89750068905353348869486981698, 7.76503337706282301303255708135, 8.473856808207040397847572203714, 9.688382657418484939686701464996, 10.96064628755246284497915381507, 11.83309635680555653509490611071
