L(s) = 1 | + (−0.596 + 5.16i)3-s + 18.4·5-s + (−4.17 + 18.0i)7-s + (−26.2 − 6.15i)9-s + 50.5i·11-s + (68.1 − 39.3i)13-s + (−11.0 + 95.3i)15-s + (−2.61 − 4.52i)17-s + (−23.3 − 13.5i)19-s + (−90.6 − 32.3i)21-s + 58.2i·23-s + 216.·25-s + (47.4 − 132. i)27-s + (−133. − 76.9i)29-s + (−1.78 − 1.03i)31-s + ⋯ |
L(s) = 1 | + (−0.114 + 0.993i)3-s + 1.65·5-s + (−0.225 + 0.974i)7-s + (−0.973 − 0.227i)9-s + 1.38i·11-s + (1.45 − 0.839i)13-s + (−0.189 + 1.64i)15-s + (−0.0372 − 0.0645i)17-s + (−0.282 − 0.163i)19-s + (−0.941 − 0.335i)21-s + 0.527i·23-s + 1.73·25-s + (0.338 − 0.941i)27-s + (−0.853 − 0.492i)29-s + (−0.0103 − 0.00596i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.284 - 0.958i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.157594815\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.157594815\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.596 - 5.16i)T \) |
| 7 | \( 1 + (4.17 - 18.0i)T \) |
good | 5 | \( 1 - 18.4T + 125T^{2} \) |
| 11 | \( 1 - 50.5iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (-68.1 + 39.3i)T + (1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (2.61 + 4.52i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (23.3 + 13.5i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 - 58.2iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (133. + 76.9i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (1.78 + 1.03i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (184. - 319. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-199. - 345. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (152. - 263. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (147. + 255. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-431. + 249. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-78.9 + 136. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (628. - 363. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-82.2 + 142. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 157. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-908. + 524. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (99.3 + 172. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-492. + 853. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (45.0 - 78.0i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-785. - 453. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−11.77918356372328895078582717779, −10.66212283097959380230456636048, −9.827502702106417610137501832093, −9.335917424883864797289596153236, −8.334932178202821078111216408171, −6.42576190299005994969458438817, −5.72829176658262015237532722482, −4.84169347522354041393044961213, −3.16423062342732437615902364531, −1.86828930504038538068438558933,
0.898211415882512556603282008343, 2.01969117019063048100524735361, 3.61562930473202841989976559244, 5.61099170304459449423936071621, 6.22141803181239223793198761868, 7.05898697134968519066906266269, 8.521039744436306913284685873464, 9.207355635461876007036703948433, 10.68320178827736643150803314852, 11.02543080760954848075082644648