| 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
−11.24379115335998450672332221883, −10.86511194125447947015160218146, −9.472480019117951824719638455338, −8.187759161100991440820562414547, −6.68671159374668190491679066537, −6.48434387096986899481884438219, −4.38292182754829194052219401580, −3.53344013943065343153004623555, −2.50769658035534828151009876239, −0.34045882222410375133578068780,
1.28141246841454595164513396188, 3.48373326716381710949614243165, 4.41879175393466705727359797031, 5.37475625861996403484675363077, 6.70164667653644681217738219101, 8.039441342283047062945468827669, 8.521435468893563888104343571167, 9.754216059467274609764665178423, 11.32766205485223941597323884253, 12.09577184854401034996592235288
