| L(s) = 1 | + (−9.79 + 5.65i)2-s + (63.9 − 110. i)4-s + (731. − 422. i)5-s + (−647. + 2.31e3i)7-s + 1.44e3i·8-s + (−4.77e3 + 8.27e3i)10-s + (−1.46e3 − 848. i)11-s − 1.20e4·13-s + (−6.73e3 − 2.63e4i)14-s + (−8.19e3 − 1.41e4i)16-s + (−4.83e4 − 2.79e4i)17-s + (−5.10e4 − 8.83e4i)19-s − 1.08e5i·20-s + 1.91e4·22-s + (−1.33e5 + 7.71e4i)23-s + ⋯ |
| L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (1.17 − 0.675i)5-s + (−0.269 + 0.962i)7-s + 0.353i·8-s + (−0.477 + 0.827i)10-s + (−0.100 − 0.0579i)11-s − 0.421·13-s + (−0.175 − 0.685i)14-s + (−0.125 − 0.216i)16-s + (−0.578 − 0.334i)17-s + (−0.391 − 0.678i)19-s − 0.675i·20-s + 0.0819·22-s + (−0.477 + 0.275i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0952i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.995 + 0.0952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.1511105709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1511105709\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (9.79 - 5.65i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (647. - 2.31e3i)T \) |
| good | 5 | \( 1 + (-731. + 422. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 11 | \( 1 + (1.46e3 + 848. i)T + (1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 + 1.20e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + (4.83e4 + 2.79e4i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (5.10e4 + 8.83e4i)T + (-8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (1.33e5 - 7.71e4i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 - 1.12e6iT - 5.00e11T^{2} \) |
| 31 | \( 1 + (-2.65e5 + 4.59e5i)T + (-4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + (1.65e5 + 2.86e5i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 - 1.44e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 9.41e5T + 1.16e13T^{2} \) |
| 47 | \( 1 + (-3.11e5 + 1.79e5i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (3.84e6 + 2.21e6i)T + (3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (8.10e6 + 4.68e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (4.37e6 + 7.58e6i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-8.83e6 + 1.52e7i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 - 4.61e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + (1.80e7 - 3.12e7i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (2.53e7 + 4.39e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + 3.31e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (7.90e7 - 4.56e7i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 + 1.64e8T + 7.83e15T^{2} \) |
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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
−12.44724218995845386975559971673, −11.17910723188154194982615574814, −9.864187560079059661938603589054, −9.216001797961073023653301904372, −8.407816462035085732974175812131, −6.84608430956025236903192141407, −5.79958521873744582991578346902, −4.92080001575682615726903957472, −2.62900191268382372995385641278, −1.56209233989587979400679874165,
0.04467276902145594447701849413, 1.62466511601786939896083371604, 2.71048146831403769290360253396, 4.18723804129220005334203572291, 6.02279258354485302139047084011, 6.91302751143304511258444389539, 8.106126269676474712716809698973, 9.537890353667829052027881819899, 10.21459159357168226608896261055, 10.86907040231359683965412614108
