| L(s) = 1 | + (−6.65 − 9.15i)2-s + (−20.1 − 62.1i)3-s + (−39.5 + 121. i)4-s + (927. + 673. i)5-s + (−434. + 597. i)6-s + (320. + 104. i)7-s + (1.37e3 − 447. i)8-s + (1.85e3 − 1.34e3i)9-s − 1.29e4i·10-s + (−6.88e3 − 1.29e4i)11-s + 8.36e3·12-s + (1.57e4 + 2.16e4i)13-s + (−1.17e3 − 3.63e3i)14-s + (2.31e4 − 7.11e4i)15-s + (−1.32e4 − 9.63e3i)16-s + (5.96e4 − 8.20e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.415 − 0.572i)2-s + (−0.249 − 0.766i)3-s + (−0.154 + 0.475i)4-s + (1.48 + 1.07i)5-s + (−0.335 + 0.461i)6-s + (0.133 + 0.0434i)7-s + (0.336 − 0.109i)8-s + (0.282 − 0.205i)9-s − 1.29i·10-s + (−0.470 − 0.882i)11-s + 0.403·12-s + (0.551 + 0.758i)13-s + (−0.0307 − 0.0945i)14-s + (0.456 − 1.40i)15-s + (−0.202 − 0.146i)16-s + (0.713 − 0.982i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.46493 - 0.811311i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46493 - 0.811311i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (6.65 + 9.15i)T \) |
| 11 | \( 1 + (6.88e3 + 1.29e4i)T \) |
| good | 3 | \( 1 + (20.1 + 62.1i)T + (-5.30e3 + 3.85e3i)T^{2} \) |
| 5 | \( 1 + (-927. - 673. i)T + (1.20e5 + 3.71e5i)T^{2} \) |
| 7 | \( 1 + (-320. - 104. i)T + (4.66e6 + 3.38e6i)T^{2} \) |
| 13 | \( 1 + (-1.57e4 - 2.16e4i)T + (-2.52e8 + 7.75e8i)T^{2} \) |
| 17 | \( 1 + (-5.96e4 + 8.20e4i)T + (-2.15e9 - 6.63e9i)T^{2} \) |
| 19 | \( 1 + (-1.33e5 + 4.33e4i)T + (1.37e10 - 9.98e9i)T^{2} \) |
| 23 | \( 1 + 1.68e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + (-9.04e5 - 2.93e5i)T + (4.04e11 + 2.94e11i)T^{2} \) |
| 31 | \( 1 + (6.73e4 - 4.89e4i)T + (2.63e11 - 8.11e11i)T^{2} \) |
| 37 | \( 1 + (-5.84e5 + 1.80e6i)T + (-2.84e12 - 2.06e12i)T^{2} \) |
| 41 | \( 1 + (-1.96e5 + 6.39e4i)T + (6.45e12 - 4.69e12i)T^{2} \) |
| 43 | \( 1 + 1.81e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (-2.51e6 - 7.73e6i)T + (-1.92e13 + 1.39e13i)T^{2} \) |
| 53 | \( 1 + (-3.73e6 + 2.71e6i)T + (1.92e13 - 5.92e13i)T^{2} \) |
| 59 | \( 1 + (5.99e6 - 1.84e7i)T + (-1.18e14 - 8.63e13i)T^{2} \) |
| 61 | \( 1 + (1.32e7 - 1.81e7i)T + (-5.92e13 - 1.82e14i)T^{2} \) |
| 67 | \( 1 + 1.13e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + (1.35e7 + 9.87e6i)T + (1.99e14 + 6.14e14i)T^{2} \) |
| 73 | \( 1 + (1.64e7 + 5.34e6i)T + (6.52e14 + 4.74e14i)T^{2} \) |
| 79 | \( 1 + (2.33e6 + 3.21e6i)T + (-4.68e14 + 1.44e15i)T^{2} \) |
| 83 | \( 1 + (-2.20e7 + 3.03e7i)T + (-6.95e14 - 2.14e15i)T^{2} \) |
| 89 | \( 1 + 8.42e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + (9.07e7 - 6.59e7i)T + (2.42e15 - 7.45e15i)T^{2} \) |
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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
−16.25226979505818392090827553131, −14.09501695120055209664493560650, −13.45561646421444091196401661762, −11.82184687639360126076666673967, −10.53528185407726573348796581639, −9.333157392095393479682255655380, −7.25583884552741398834577331146, −5.93333043236514544405297873184, −2.81623387417189518239981087772, −1.26305258498159436406643236682,
1.44746202224236928594293473071, 4.82533639629643498140182202295, 5.86130277960471993647066868813, 8.138632354079640348468953162099, 9.746065914225593387743905118780, 10.25931369155312196534132158509, 12.67307965930112876819241335364, 13.84681372719830464618191730905, 15.43450160884218773437364144321, 16.46219758168483380481904085447
