| L(s) = 1 | + (−1 + 1.73i)2-s + (1.5 + 2.59i)3-s + (−1.99 − 3.46i)4-s + (−0.436 + 0.756i)5-s − 6·6-s + (−10.8 − 15.0i)7-s + 7.99·8-s + (−4.5 + 7.79i)9-s + (−0.872 − 1.51i)10-s + (0.682 + 1.18i)11-s + (6.00 − 10.3i)12-s + 13·13-s + (36.8 − 3.68i)14-s − 2.61·15-s + (−8 + 13.8i)16-s + (−12.5 − 21.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.0390 + 0.0676i)5-s − 0.408·6-s + (−0.583 − 0.812i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.0276 − 0.0478i)10-s + (0.0187 + 0.0324i)11-s + (0.144 − 0.249i)12-s + 0.277·13-s + (0.703 − 0.0703i)14-s − 0.0450·15-s + (−0.125 + 0.216i)16-s + (−0.179 − 0.310i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.162 - 0.986i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.405191850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.405191850\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1 - 1.73i)T \) |
| 3 | \( 1 + (-1.5 - 2.59i)T \) |
| 7 | \( 1 + (10.8 + 15.0i)T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 + (0.436 - 0.756i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-0.682 - 1.18i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 17 | \( 1 + (12.5 + 21.7i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (6.26 - 10.8i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-39.5 + 68.4i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 3.98T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-97.5 - 168. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (90.7 - 157. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 408.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 295.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (178. - 308. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-181. - 314. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (155. + 269. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-121. + 210. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-128. - 222. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 75.4T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-127. - 221. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (27.8 - 48.2i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 33.5T + 5.71e5T^{2} \) |
| 89 | \( 1 + (551. - 954. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.33e3T + 9.12e5T^{2} \) |
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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
−10.53768513450796724013311336627, −9.609181128078759009324877207299, −8.954486016476252462397557132593, −7.950291712212466303708093993510, −7.06160685122303986387054534099, −6.25271310595312942006673861860, −5.00794801752937970532170675883, −4.04503673473321020757465147777, −2.88761711938818788871091813228, −0.996986323588825374815426125722,
0.57928545840376412565383915652, 2.06140689327859446780712541897, 2.97546712142729630976411463028, 4.14512384704835108592570632680, 5.60827157513150352206445087079, 6.55537984449411162856489134116, 7.63895585955155448924392158714, 8.592569714862271475051351864291, 9.181616230951843815969457151918, 10.07170109614509293270217168625
