| L(s) = 1 | + (1.68 + 1.08i)2-s + (0.0867 + 0.603i)3-s + (1.66 + 3.63i)4-s + (−4.79 − 1.40i)5-s + (−0.506 + 1.10i)6-s + (−3.02 + 3.49i)7-s + (−1.13 + 7.91i)8-s + (25.5 − 7.50i)9-s + (−6.54 − 7.55i)10-s + (−47.2 + 30.3i)11-s + (−2.05 + 1.31i)12-s + (40.8 + 47.1i)13-s + (−8.87 + 2.60i)14-s + (0.433 − 3.01i)15-s + (−10.4 + 12.0i)16-s + (−45.2 + 99.0i)17-s + ⋯ |
| L(s) = 1 | + (0.594 + 0.382i)2-s + (0.0166 + 0.116i)3-s + (0.207 + 0.454i)4-s + (−0.429 − 0.125i)5-s + (−0.0344 + 0.0754i)6-s + (−0.163 + 0.188i)7-s + (−0.0503 + 0.349i)8-s + (0.946 − 0.277i)9-s + (−0.207 − 0.238i)10-s + (−1.29 + 0.833i)11-s + (−0.0493 + 0.0317i)12-s + (0.871 + 1.00i)13-s + (−0.169 + 0.0497i)14-s + (0.00746 − 0.0519i)15-s + (−0.163 + 0.188i)16-s + (−0.645 + 1.41i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.628 - 0.777i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.628 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.778888 + 1.63071i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.778888 + 1.63071i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.68 - 1.08i)T \) |
| 5 | \( 1 + (4.79 + 1.40i)T \) |
| 23 | \( 1 + (109. + 8.25i)T \) |
| good | 3 | \( 1 + (-0.0867 - 0.603i)T + (-25.9 + 7.60i)T^{2} \) |
| 7 | \( 1 + (3.02 - 3.49i)T + (-48.8 - 339. i)T^{2} \) |
| 11 | \( 1 + (47.2 - 30.3i)T + (552. - 1.21e3i)T^{2} \) |
| 13 | \( 1 + (-40.8 - 47.1i)T + (-312. + 2.17e3i)T^{2} \) |
| 17 | \( 1 + (45.2 - 99.0i)T + (-3.21e3 - 3.71e3i)T^{2} \) |
| 19 | \( 1 + (5.84 + 12.8i)T + (-4.49e3 + 5.18e3i)T^{2} \) |
| 29 | \( 1 + (71.0 - 155. i)T + (-1.59e4 - 1.84e4i)T^{2} \) |
| 31 | \( 1 + (-10.9 + 75.8i)T + (-2.85e4 - 8.39e3i)T^{2} \) |
| 37 | \( 1 + (-145. + 42.6i)T + (4.26e4 - 2.73e4i)T^{2} \) |
| 41 | \( 1 + (-31.5 - 9.25i)T + (5.79e4 + 3.72e4i)T^{2} \) |
| 43 | \( 1 + (32.3 + 224. i)T + (-7.62e4 + 2.23e4i)T^{2} \) |
| 47 | \( 1 - 357.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (165. - 191. i)T + (-2.11e4 - 1.47e5i)T^{2} \) |
| 59 | \( 1 + (60.7 + 70.1i)T + (-2.92e4 + 2.03e5i)T^{2} \) |
| 61 | \( 1 + (-61.9 + 430. i)T + (-2.17e5 - 6.39e4i)T^{2} \) |
| 67 | \( 1 + (-255. - 164. i)T + (1.24e5 + 2.73e5i)T^{2} \) |
| 71 | \( 1 + (-511. - 329. i)T + (1.48e5 + 3.25e5i)T^{2} \) |
| 73 | \( 1 + (-104. - 228. i)T + (-2.54e5 + 2.93e5i)T^{2} \) |
| 79 | \( 1 + (224. + 258. i)T + (-7.01e4 + 4.88e5i)T^{2} \) |
| 83 | \( 1 + (-655. + 192. i)T + (4.81e5 - 3.09e5i)T^{2} \) |
| 89 | \( 1 + (99.1 + 689. i)T + (-6.76e5 + 1.98e5i)T^{2} \) |
| 97 | \( 1 + (694. + 203. i)T + (7.67e5 + 4.93e5i)T^{2} \) |
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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
−12.52141212061671527492811080161, −11.19502055638404300797988267545, −10.29140786275722745971318423912, −9.059331962037197352782220654955, −7.974770957756537479937825930374, −7.00708687269995078277495486577, −5.97164596039721376463105176627, −4.56624946181932224303624271212, −3.82292441234174538914729168414, −1.99856497837710358092126818425,
0.59749265874028215892712682314, 2.54382479625917909705679097337, 3.74714354863656110318413277053, 4.98586492608058971872893112099, 6.11188410577206948958088455375, 7.42280192090204966122319077584, 8.244836222761822934916474735199, 9.790143108269653545849471406032, 10.65214781285646443361487388195, 11.35204844655819025526536938804
