| L(s) = 1 | + (0.0871 − 0.996i)2-s + (−0.984 − 0.173i)4-s + (−2.82 − 1.31i)5-s + (1.97 − 0.718i)7-s + (−0.258 + 0.965i)8-s + (−1.55 + 2.70i)10-s + (−1.33 − 2.31i)11-s + (−0.527 − 0.753i)13-s + (−0.543 − 2.02i)14-s + (0.939 + 0.342i)16-s + (−2.87 + 4.11i)17-s + (−2.92 + 0.256i)19-s + (2.55 + 1.78i)20-s + (−2.42 + 1.13i)22-s + (−2.70 + 0.725i)23-s + ⋯ |
| L(s) = 1 | + (0.0616 − 0.704i)2-s + (−0.492 − 0.0868i)4-s + (−1.26 − 0.589i)5-s + (0.745 − 0.271i)7-s + (−0.0915 + 0.341i)8-s + (−0.493 + 0.853i)10-s + (−0.403 − 0.699i)11-s + (−0.146 − 0.208i)13-s + (−0.145 − 0.541i)14-s + (0.234 + 0.0855i)16-s + (−0.698 + 0.997i)17-s + (−0.671 + 0.0587i)19-s + (0.571 + 0.399i)20-s + (−0.517 + 0.241i)22-s + (−0.564 + 0.151i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.740 - 0.672i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.740 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.117194 + 0.303201i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.117194 + 0.303201i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0871 + 0.996i)T \) |
| 3 | \( 1 \) |
| 37 | \( 1 + (1.40 - 5.91i)T \) |
| good | 5 | \( 1 + (2.82 + 1.31i)T + (3.21 + 3.83i)T^{2} \) |
| 7 | \( 1 + (-1.97 + 0.718i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (1.33 + 2.31i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.527 + 0.753i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (2.87 - 4.11i)T + (-5.81 - 15.9i)T^{2} \) |
| 19 | \( 1 + (2.92 - 0.256i)T + (18.7 - 3.29i)T^{2} \) |
| 23 | \( 1 + (2.70 - 0.725i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (4.95 + 1.32i)T + (25.1 + 14.5i)T^{2} \) |
| 31 | \( 1 + (1.88 + 1.88i)T + 31iT^{2} \) |
| 41 | \( 1 + (0.366 - 2.08i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (6.00 - 6.00i)T - 43iT^{2} \) |
| 47 | \( 1 + (-5.03 - 2.90i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.29 + 6.30i)T + (-40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (5.91 + 12.6i)T + (-37.9 + 45.1i)T^{2} \) |
| 61 | \( 1 + (-5.22 + 3.66i)T + (20.8 - 57.3i)T^{2} \) |
| 67 | \( 1 + (2.91 + 8.01i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-3.95 + 4.70i)T + (-12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + 9.01iT - 73T^{2} \) |
| 79 | \( 1 + (-0.222 + 0.477i)T + (-50.7 - 60.5i)T^{2} \) |
| 83 | \( 1 + (3.14 - 0.553i)T + (77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-2.30 + 1.07i)T + (57.2 - 68.1i)T^{2} \) |
| 97 | \( 1 + (3.13 + 11.7i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−10.21457818122661954884968154619, −9.022155441638639541422498336796, −8.082194113121738774390983175525, −7.916358678979085974837991554957, −6.31552993813819425447814294781, −5.03320230967826684259183378393, −4.25446504996079934754741999424, −3.41975604955462289404149030773, −1.79398027183848093873137138886, −0.16643199921917436885968362401,
2.33196521301879739669824707380, 3.83421047545505717167619936392, 4.62058149961256132880186097259, 5.61120163360672872123298717591, 7.12846411130170043832982953812, 7.24556001745763448160219669676, 8.320724998767581701447164495620, 9.017495721885232813874317800747, 10.25217033407604936111615223508, 11.12620735763390057214970699123
