| L(s) = 1 | + (−0.437 + 0.0692i)2-s + (−2.82 − 1.44i)3-s + (−1.71 + 0.557i)4-s + (2.19 − 0.426i)5-s + (1.33 + 0.434i)6-s + (−2.15 + 1.52i)7-s + (1.50 − 0.764i)8-s + (4.15 + 5.72i)9-s + (−0.930 + 0.338i)10-s + (2.41 + 1.75i)11-s + (5.65 + 0.895i)12-s + (−0.482 + 3.04i)13-s + (0.838 − 0.817i)14-s + (−6.82 − 1.95i)15-s + (2.31 − 1.68i)16-s + (−0.763 + 0.388i)17-s + ⋯ |
| L(s) = 1 | + (−0.309 + 0.0489i)2-s + (−1.63 − 0.831i)3-s + (−0.857 + 0.278i)4-s + (0.981 − 0.190i)5-s + (0.545 + 0.177i)6-s + (−0.816 + 0.577i)7-s + (0.530 − 0.270i)8-s + (1.38 + 1.90i)9-s + (−0.294 + 0.107i)10-s + (0.729 + 0.529i)11-s + (1.63 + 0.258i)12-s + (−0.133 + 0.844i)13-s + (0.224 − 0.218i)14-s + (−1.76 − 0.505i)15-s + (0.578 − 0.420i)16-s + (−0.185 + 0.0942i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.373 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.373 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.350028 + 0.236400i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.350028 + 0.236400i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.19 + 0.426i)T \) |
| 7 | \( 1 + (2.15 - 1.52i)T \) |
| good | 2 | \( 1 + (0.437 - 0.0692i)T + (1.90 - 0.618i)T^{2} \) |
| 3 | \( 1 + (2.82 + 1.44i)T + (1.76 + 2.42i)T^{2} \) |
| 11 | \( 1 + (-2.41 - 1.75i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.482 - 3.04i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (0.763 - 0.388i)T + (9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (1.72 - 5.31i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.214 - 1.35i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (4.22 - 1.37i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (5.16 + 1.67i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.429 + 2.71i)T + (-35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (-6.23 - 8.58i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-4.86 - 4.86i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.788 - 1.54i)T + (-27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (5.16 - 10.1i)T + (-31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (0.0470 - 0.0341i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (3.65 - 5.02i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (0.445 + 0.873i)T + (-39.3 + 54.2i)T^{2} \) |
| 71 | \( 1 + (2.53 + 7.78i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-4.88 + 0.774i)T + (69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (-8.47 + 2.75i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.465 - 0.914i)T + (-48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (8.07 + 5.86i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (0.309 - 0.607i)T + (-57.0 - 78.4i)T^{2} \) |
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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
−12.63869475854658588016455645769, −12.28255392554960213760025250214, −10.97276743225161841114817362581, −9.777353241876585691310085343143, −9.159993138618974808400049198276, −7.51141317568750070427704676783, −6.36955265686510066095523449240, −5.70023822275896722419752257940, −4.41787252572910927235503608605, −1.62219928648560332229363161001,
0.56708846416415633010449377965, 3.84331333760395699962079724520, 5.10134309652883417245776180949, 5.92523583542380602622572518674, 6.88715188720925828832112744609, 9.087274570084817914418519194755, 9.665693236170950595113422531479, 10.58151949227458113717411404981, 11.03028439245353963590529007719, 12.59501759652911214230370206428
