| L(s) = 1 | + (1.48 + 0.0777i)2-s + (2.51 + 3.10i)3-s + (−5.76 − 0.605i)4-s + (4.25 + 10.3i)5-s + (3.48 + 4.79i)6-s + (−12.4 + 13.7i)7-s + (−20.2 − 3.20i)8-s + (2.30 − 10.8i)9-s + (5.50 + 15.6i)10-s + (−33.2 + 7.06i)11-s + (−12.5 − 19.3i)12-s + (−13.9 − 7.08i)13-s + (−19.4 + 19.4i)14-s + (−21.3 + 39.1i)15-s + (15.5 + 3.30i)16-s + (−27.0 − 70.3i)17-s + ⋯ |
| L(s) = 1 | + (0.524 + 0.0274i)2-s + (0.483 + 0.596i)3-s + (−0.720 − 0.0756i)4-s + (0.380 + 0.924i)5-s + (0.237 + 0.326i)6-s + (−0.670 + 0.741i)7-s + (−0.894 − 0.141i)8-s + (0.0853 − 0.401i)9-s + (0.174 + 0.495i)10-s + (−0.911 + 0.193i)11-s + (−0.302 − 0.466i)12-s + (−0.296 − 0.151i)13-s + (−0.372 + 0.370i)14-s + (−0.368 + 0.673i)15-s + (0.242 + 0.0516i)16-s + (−0.385 − 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 - 0.346i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.938 - 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.210757 + 1.17916i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.210757 + 1.17916i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-4.25 - 10.3i)T \) |
| 7 | \( 1 + (12.4 - 13.7i)T \) |
| good | 2 | \( 1 + (-1.48 - 0.0777i)T + (7.95 + 0.836i)T^{2} \) |
| 3 | \( 1 + (-2.51 - 3.10i)T + (-5.61 + 26.4i)T^{2} \) |
| 11 | \( 1 + (33.2 - 7.06i)T + (1.21e3 - 541. i)T^{2} \) |
| 13 | \( 1 + (13.9 + 7.08i)T + (1.29e3 + 1.77e3i)T^{2} \) |
| 17 | \( 1 + (27.0 + 70.3i)T + (-3.65e3 + 3.28e3i)T^{2} \) |
| 19 | \( 1 + (-3.94 - 37.5i)T + (-6.70e3 + 1.42e3i)T^{2} \) |
| 23 | \( 1 + (10.3 - 197. i)T + (-1.21e4 - 1.27e3i)T^{2} \) |
| 29 | \( 1 + (33.8 - 46.6i)T + (-7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (69.5 - 156. i)T + (-1.99e4 - 2.21e4i)T^{2} \) |
| 37 | \( 1 + (-48.0 + 31.2i)T + (2.06e4 - 4.62e4i)T^{2} \) |
| 41 | \( 1 + (17.8 - 5.78i)T + (5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + (-1.93 - 1.93i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + (-535. - 205. i)T + (7.71e4 + 6.94e4i)T^{2} \) |
| 53 | \( 1 + (334. - 271. i)T + (3.09e4 - 1.45e5i)T^{2} \) |
| 59 | \( 1 + (215. - 239. i)T + (-2.14e4 - 2.04e5i)T^{2} \) |
| 61 | \( 1 + (355. - 320. i)T + (2.37e4 - 2.25e5i)T^{2} \) |
| 67 | \( 1 + (-720. + 276. i)T + (2.23e5 - 2.01e5i)T^{2} \) |
| 71 | \( 1 + (-617. - 448. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-20.1 + 31.0i)T + (-1.58e5 - 3.55e5i)T^{2} \) |
| 79 | \( 1 + (294. + 662. i)T + (-3.29e5 + 3.66e5i)T^{2} \) |
| 83 | \( 1 + (-227. + 1.43e3i)T + (-5.43e5 - 1.76e5i)T^{2} \) |
| 89 | \( 1 + (-734. - 815. i)T + (-7.36e4 + 7.01e5i)T^{2} \) |
| 97 | \( 1 + (96.4 + 609. i)T + (-8.68e5 + 2.82e5i)T^{2} \) |
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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
−12.86607757279030648157196596565, −11.90497349934201635051983916792, −10.44593757171223682120939741160, −9.545685993872558889789129980679, −9.081886042521281844962145011983, −7.46422780552423942415803353876, −6.07890690616351800224560330341, −5.12230524558620231194843678044, −3.58297326273178639546240882712, −2.77000840651938107780579127008,
0.42174017673805608260217994866, 2.41383458604246806086513556907, 4.07896833964963224778327328824, 5.08549892155989603016269598277, 6.38286201754586518319478229837, 7.86438447423932726051754584456, 8.619892479278414319372452562387, 9.693995008624657875305805904550, 10.73149232516426900647403863546, 12.58069227741889351474484443867
