| L(s) = 1 | + (2.17 − 2.31i)2-s + (−4.67 + 1.47i)3-s + (−0.391 − 6.09i)4-s + (−6.86 + 3.72i)5-s + (−6.76 + 14.0i)6-s + (12.8 + 0.411i)7-s + (−5.15 − 4.25i)8-s + (12.3 − 8.61i)9-s + (−6.29 + 24.0i)10-s + (2.17 + 14.9i)11-s + (10.7 + 27.9i)12-s + (−0.0196 − 1.22i)13-s + (28.8 − 28.8i)14-s + (26.6 − 27.5i)15-s + (3.13 − 0.404i)16-s + (−11.4 + 27.0i)17-s + ⋯ |
| L(s) = 1 | + (1.08 − 1.15i)2-s + (−1.55 + 0.490i)3-s + (−0.0977 − 1.52i)4-s + (−1.37 + 0.744i)5-s + (−1.12 + 2.34i)6-s + (1.83 + 0.0587i)7-s + (−0.644 − 0.531i)8-s + (1.37 − 0.957i)9-s + (−0.629 + 2.40i)10-s + (0.198 + 1.36i)11-s + (0.899 + 2.32i)12-s + (−0.00151 − 0.0943i)13-s + (2.05 − 2.05i)14-s + (1.77 − 1.83i)15-s + (0.196 − 0.0252i)16-s + (−0.673 + 1.59i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.414i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.909 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.29781 + 0.282006i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29781 + 0.282006i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 + (-162. + 110. i)T \) |
| good | 2 | \( 1 + (-2.17 + 2.31i)T + (-0.256 - 3.99i)T^{2} \) |
| 3 | \( 1 + (4.67 - 1.47i)T + (7.38 - 5.14i)T^{2} \) |
| 5 | \( 1 + (6.86 - 3.72i)T + (13.6 - 20.9i)T^{2} \) |
| 7 | \( 1 + (-12.8 - 0.411i)T + (48.8 + 3.13i)T^{2} \) |
| 11 | \( 1 + (-2.17 - 14.9i)T + (-115. + 34.4i)T^{2} \) |
| 13 | \( 1 + (0.0196 + 1.22i)T + (-168. + 5.41i)T^{2} \) |
| 17 | \( 1 + (11.4 - 27.0i)T + (-201. - 207. i)T^{2} \) |
| 19 | \( 1 + (10.0 - 2.28i)T + (325. - 156. i)T^{2} \) |
| 23 | \( 1 + (7.42 - 6.32i)T + (84.4 - 522. i)T^{2} \) |
| 29 | \( 1 + (8.56 - 19.3i)T + (-565. - 622. i)T^{2} \) |
| 31 | \( 1 + (-14.7 - 21.9i)T + (-360. + 890. i)T^{2} \) |
| 37 | \( 1 + (-1.22 - 0.157i)T + (1.32e3 + 347. i)T^{2} \) |
| 41 | \( 1 + (19.7 + 48.6i)T + (-1.20e3 + 1.16e3i)T^{2} \) |
| 43 | \( 1 + (17.3 - 23.2i)T + (-526. - 1.77e3i)T^{2} \) |
| 47 | \( 1 + (-25.6 + 15.5i)T + (1.02e3 - 1.95e3i)T^{2} \) |
| 53 | \( 1 + (-10.8 - 11.9i)T + (-269. + 2.79e3i)T^{2} \) |
| 59 | \( 1 + (-11.5 + 3.02i)T + (3.03e3 - 1.70e3i)T^{2} \) |
| 61 | \( 1 + (-6.56 - 12.5i)T + (-2.12e3 + 3.05e3i)T^{2} \) |
| 67 | \( 1 + (-15.6 + 63.7i)T + (-3.97e3 - 2.07e3i)T^{2} \) |
| 71 | \( 1 + (-22.1 - 3.94i)T + (4.73e3 + 1.74e3i)T^{2} \) |
| 73 | \( 1 + (-81.1 - 105. i)T + (-1.35e3 + 5.15e3i)T^{2} \) |
| 79 | \( 1 + (88.6 + 48.0i)T + (3.40e3 + 5.23e3i)T^{2} \) |
| 83 | \( 1 + (50.7 + 11.5i)T + (6.20e3 + 2.98e3i)T^{2} \) |
| 89 | \( 1 + (5.98 - 8.88i)T + (-2.97e3 - 7.34e3i)T^{2} \) |
| 97 | \( 1 + (-2.00 + 3.55i)T + (-4.87e3 - 8.04e3i)T^{2} \) |
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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
−12.04000467788687734248907905543, −11.44659973623595327454406544074, −10.77974752067222719732108815178, −10.38763669359634762668683343159, −8.169552593564650504430545403572, −6.90498440822241386581601040753, −5.40903015624125343443189507572, −4.44590670287547514772666105818, −4.02012012683585116068198532383, −1.75851899648215117344723203091,
0.70786113244743231284907217080, 4.25745698001467814545430645700, 4.83439046462888420290273260554, 5.66610741674244562181867258615, 6.85520503756268188472157880148, 7.80759738833150912182303255652, 8.458422133818373171839458665940, 11.03672226779028858046544385884, 11.57537213089328573488469392717, 12.01980381480298488417054656237
