| L(s) = 1 | + (1.07 + 0.0941i)2-s + (5.17 − 0.517i)3-s + (−6.72 − 1.18i)4-s + (9.59 + 5.73i)5-s + (5.61 − 0.0704i)6-s + (14.8 + 10.3i)7-s + (−15.4 − 4.14i)8-s + (26.4 − 5.35i)9-s + (9.79 + 7.07i)10-s + (−2.20 + 6.06i)11-s + (−35.4 − 2.65i)12-s + (1.13 + 12.9i)13-s + (14.9 + 12.5i)14-s + (52.5 + 24.6i)15-s + (35.0 + 12.7i)16-s + (111. − 29.7i)17-s + ⋯ |
| L(s) = 1 | + (0.380 + 0.0332i)2-s + (0.995 − 0.0996i)3-s + (−0.841 − 0.148i)4-s + (0.858 + 0.512i)5-s + (0.381 − 0.00479i)6-s + (0.800 + 0.560i)7-s + (−0.684 − 0.183i)8-s + (0.980 − 0.198i)9-s + (0.309 + 0.223i)10-s + (−0.0604 + 0.166i)11-s + (−0.851 − 0.0637i)12-s + (0.0241 + 0.275i)13-s + (0.286 + 0.240i)14-s + (0.905 + 0.424i)15-s + (0.548 + 0.199i)16-s + (1.58 − 0.424i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.307i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.951 - 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.71282 + 0.428040i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.71282 + 0.428040i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-5.17 + 0.517i)T \) |
| 5 | \( 1 + (-9.59 - 5.73i)T \) |
| good | 2 | \( 1 + (-1.07 - 0.0941i)T + (7.87 + 1.38i)T^{2} \) |
| 7 | \( 1 + (-14.8 - 10.3i)T + (117. + 322. i)T^{2} \) |
| 11 | \( 1 + (2.20 - 6.06i)T + (-1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (-1.13 - 12.9i)T + (-2.16e3 + 381. i)T^{2} \) |
| 17 | \( 1 + (-111. + 29.7i)T + (4.25e3 - 2.45e3i)T^{2} \) |
| 19 | \( 1 + (28.5 - 16.4i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (87.8 + 125. i)T + (-4.16e3 + 1.14e4i)T^{2} \) |
| 29 | \( 1 + (160. - 134. i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-6.38 + 36.2i)T + (-2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (86.2 + 322. i)T + (-4.38e4 + 2.53e4i)T^{2} \) |
| 41 | \( 1 + (154. - 184. i)T + (-1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (69.5 + 149. i)T + (-5.11e4 + 6.09e4i)T^{2} \) |
| 47 | \( 1 + (100. - 143. i)T + (-3.55e4 - 9.75e4i)T^{2} \) |
| 53 | \( 1 + (178. - 178. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (415. - 151. i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (126. + 719. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (777. - 68.0i)T + (2.96e5 - 5.22e4i)T^{2} \) |
| 71 | \( 1 + (160. + 92.8i)T + (1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-33.0 + 123. i)T + (-3.36e5 - 1.94e5i)T^{2} \) |
| 79 | \( 1 + (-2.86 - 3.41i)T + (-8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-78.4 + 896. i)T + (-5.63e5 - 9.92e4i)T^{2} \) |
| 89 | \( 1 + (-176. - 305. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-83.1 + 38.7i)T + (5.86e5 - 6.99e5i)T^{2} \) |
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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
−13.00756090026146548607537254300, −12.16873387120977802712090689245, −10.47012460255966960580545528875, −9.552606316198259548942873836261, −8.722267003054189320472455915713, −7.60084516841704652075674608141, −6.01533942015324179951898687401, −4.83933543322691818008764698883, −3.33920840703825732907546821557, −1.83087528294467511412781083910,
1.48974706760253337369437384406, 3.40376638236257751478342609948, 4.61120940457286890409312763615, 5.71424716395390779541894376853, 7.73476107678879576922540700201, 8.449566750413762545695799011917, 9.571570657331587794026257716084, 10.26819499225396158687180773148, 12.02404317629429626839708615051, 13.12744675490099931439131683074
