L(s) = 1 | − 1.67·2-s + (−0.5 − 0.866i)3-s + 0.806·4-s + (−1.64 + 2.84i)5-s + (0.837 + 1.45i)6-s + (1.07 + 1.86i)7-s + 1.99·8-s + (−0.499 + 0.866i)9-s + (2.75 − 4.76i)10-s + (−2.31 + 4.01i)11-s + (−0.403 − 0.698i)12-s + (−1.09 + 1.90i)13-s + (−1.80 − 3.12i)14-s + 3.28·15-s − 4.96·16-s + (−3.15 − 5.46i)17-s + ⋯ |
L(s) = 1 | − 1.18·2-s + (−0.288 − 0.499i)3-s + 0.403·4-s + (−0.735 + 1.27i)5-s + (0.341 + 0.592i)6-s + (0.407 + 0.705i)7-s + 0.707·8-s + (−0.166 + 0.288i)9-s + (0.870 − 1.50i)10-s + (−0.699 + 1.21i)11-s + (−0.116 − 0.201i)12-s + (−0.304 + 0.526i)13-s + (−0.482 − 0.836i)14-s + 0.848·15-s − 1.24·16-s + (−0.765 − 1.32i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0890 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0890 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.241901 + 0.264482i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.241901 + 0.264482i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (5.44 - 1.16i)T \) |
good | 2 | \( 1 + 1.67T + 2T^{2} \) |
| 5 | \( 1 + (1.64 - 2.84i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.07 - 1.86i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.31 - 4.01i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.09 - 1.90i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3.15 + 5.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.903 - 1.56i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 7.35T + 23T^{2} \) |
| 29 | \( 1 + 0.0630T + 29T^{2} \) |
| 37 | \( 1 + (-1.96 - 3.40i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.12 - 3.68i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.19 - 3.79i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 6.37T + 47T^{2} \) |
| 53 | \( 1 + (-5.99 + 10.3i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.48 - 7.76i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 - 3.19T + 61T^{2} \) |
| 67 | \( 1 + (3.79 - 6.56i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.86 - 8.43i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.94 + 8.56i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.07 + 1.86i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.64 + 6.31i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 3.86T + 89T^{2} \) |
| 97 | \( 1 + 5.38T + 97T^{2} \) |
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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
−14.56770083551896285147693327874, −13.18191969114594025088780218514, −11.75997282168686375646474981652, −11.09286622734950875179305460761, −9.979886253516688846607694999996, −8.775871702847011540309199372787, −7.38140735931465324054915847187, −7.07120500840648999171540317920, −4.87180526331226723995272116941, −2.41197146249665193781138251046,
0.66915458485712183178281875717, 4.11227704118829189618629977794, 5.29786618839077847943901229841, 7.48037944117080805241767162640, 8.459375948201960790092003590123, 9.088449414231367902712342050110, 10.65531192606758505038598142736, 11.05154690287356380871156093903, 12.66867704310149398723157159887, 13.58222298764305809221292049313