| L(s) = 1 | + (3.23 + 2.35i)2-s + (4.94 + 15.2i)4-s + (−63.1 − 86.9i)5-s + (−48.9 + 15.9i)7-s + (−19.7 + 60.8i)8-s − 429. i·10-s + (162. + 366. i)11-s + (532. − 732. i)13-s + (−195. − 63.6i)14-s + (−207. + 150. i)16-s + (−1.47e3 + 1.06e3i)17-s + (479. + 155. i)19-s + (1.01e3 − 1.39e3i)20-s + (−336. + 1.56e3i)22-s + 3.19e3i·23-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (0.154 + 0.475i)4-s + (−1.12 − 1.55i)5-s + (−0.377 + 0.122i)7-s + (−0.109 + 0.336i)8-s − 1.35i·10-s + (0.405 + 0.914i)11-s + (0.873 − 1.20i)13-s + (−0.266 − 0.0867i)14-s + (−0.202 + 0.146i)16-s + (−1.23 + 0.897i)17-s + (0.304 + 0.0990i)19-s + (0.564 − 0.777i)20-s + (−0.148 + 0.691i)22-s + 1.25i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.352 - 0.935i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.352 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.250743942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.250743942\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.23 - 2.35i)T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (-162. - 366. i)T \) |
| good | 5 | \( 1 + (63.1 + 86.9i)T + (-965. + 2.97e3i)T^{2} \) |
| 7 | \( 1 + (48.9 - 15.9i)T + (1.35e4 - 9.87e3i)T^{2} \) |
| 13 | \( 1 + (-532. + 732. i)T + (-1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (1.47e3 - 1.06e3i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-479. - 155. i)T + (2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 - 3.19e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (445. + 1.36e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-6.82e3 - 4.96e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-2.91e3 - 8.96e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (1.79e3 - 5.52e3i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 + 7.08e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (2.75e4 + 8.95e3i)T + (1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-1.77e3 + 2.43e3i)T + (-1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-1.07e4 + 3.49e3i)T + (5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-1.75e4 - 2.42e4i)T + (-2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 - 2.41e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-3.95e4 - 5.44e4i)T + (-5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (1.14e4 - 3.71e3i)T + (1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (1.51e4 - 2.08e4i)T + (-9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-4.85e4 + 3.52e4i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 - 1.11e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (4.45e4 + 3.23e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−12.09577184854401034996592235288, −11.32766205485223941597323884253, −9.754216059467274609764665178423, −8.521435468893563888104343571167, −8.039441342283047062945468827669, −6.70164667653644681217738219101, −5.37475625861996403484675363077, −4.41879175393466705727359797031, −3.48373326716381710949614243165, −1.28141246841454595164513396188,
0.34045882222410375133578068780, 2.50769658035534828151009876239, 3.53344013943065343153004623555, 4.38292182754829194052219401580, 6.48434387096986899481884438219, 6.68671159374668190491679066537, 8.187759161100991440820562414547, 9.472480019117951824719638455338, 10.86511194125447947015160218146, 11.24379115335998450672332221883
