| L(s) = 1 | + (1.29 − 0.566i)2-s + (0.755 − 0.654i)3-s + (1.35 − 1.46i)4-s + (0.907 − 0.130i)5-s + (0.608 − 1.27i)6-s + (−1.67 + 3.67i)7-s + (0.927 − 2.67i)8-s + (0.142 − 0.989i)9-s + (1.10 − 0.682i)10-s + (1.75 − 5.96i)11-s + (0.0645 − 1.99i)12-s + (−0.258 + 0.117i)13-s + (−0.0922 + 5.71i)14-s + (0.600 − 0.692i)15-s + (−0.312 − 3.98i)16-s + (3.14 + 2.02i)17-s + ⋯ |
| L(s) = 1 | + (0.916 − 0.400i)2-s + (0.436 − 0.378i)3-s + (0.678 − 0.734i)4-s + (0.405 − 0.0583i)5-s + (0.248 − 0.521i)6-s + (−0.634 + 1.39i)7-s + (0.327 − 0.944i)8-s + (0.0474 − 0.329i)9-s + (0.348 − 0.215i)10-s + (0.527 − 1.79i)11-s + (0.0186 − 0.577i)12-s + (−0.0716 + 0.0327i)13-s + (−0.0246 + 1.52i)14-s + (0.154 − 0.178i)15-s + (−0.0782 − 0.996i)16-s + (0.762 + 0.490i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.478 + 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.478 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.51540 - 1.49367i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.51540 - 1.49367i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.29 + 0.566i)T \) |
| 3 | \( 1 + (-0.755 + 0.654i)T \) |
| 23 | \( 1 + (-4.60 + 1.34i)T \) |
| good | 5 | \( 1 + (-0.907 + 0.130i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (1.67 - 3.67i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-1.75 + 5.96i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (0.258 - 0.117i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-3.14 - 2.02i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (0.0815 + 0.126i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (4.40 - 6.85i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (0.844 - 0.974i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-3.53 - 0.507i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-1.27 - 8.85i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (4.63 - 4.01i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + (2.73 + 1.24i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (13.0 - 5.95i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-8.58 - 7.43i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (2.35 + 8.01i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-8.58 + 2.52i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-2.88 + 1.85i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (0.819 + 1.79i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-14.2 - 2.05i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-4.87 - 5.62i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (1.04 + 7.28i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.93671844576405132966342648683, −9.637380193654828732868245300865, −9.053523641352326273597656741381, −8.036439674241063578743077048181, −6.52361554836659845195042689632, −6.01432237822728074505874062111, −5.19652545260933670529618422734, −3.44517752178212133018282099393, −2.92883311584019497789130203173, −1.50153994282280404982587745222,
2.03622027102785543829018473452, 3.48168493795858217030866875272, 4.21612469907478285979098482270, 5.16301873834562791139439740860, 6.49032725907424845317273930970, 7.24880122813006329582679636975, 7.84433682641454475635520101507, 9.520485895674834216352694184625, 9.872617678708479355455989924246, 10.93028108504644137747875686901
