| L(s) = 1 | + (8.85 − 15.3i)2-s + (64.7 + 124. i)3-s + (99.0 + 171. i)4-s + (369. + 640. i)5-s + (2.48e3 + 108. i)6-s + (3.01e3 − 5.21e3i)7-s + 1.25e4·8-s + (−1.12e4 + 1.61e4i)9-s + 1.31e4·10-s + (−9.21e3 + 1.59e4i)11-s + (−1.49e4 + 2.34e4i)12-s + (−5.87e4 − 1.01e5i)13-s + (−5.33e4 − 9.24e4i)14-s + (−5.57e4 + 8.75e4i)15-s + (6.07e4 − 1.05e5i)16-s − 4.68e5·17-s + ⋯ |
| L(s) = 1 | + (0.391 − 0.678i)2-s + (0.461 + 0.887i)3-s + (0.193 + 0.335i)4-s + (0.264 + 0.458i)5-s + (0.782 + 0.0342i)6-s + (0.474 − 0.821i)7-s + 1.08·8-s + (−0.573 + 0.819i)9-s + 0.414·10-s + (−0.189 + 0.328i)11-s + (−0.207 + 0.326i)12-s + (−0.570 − 0.988i)13-s + (−0.371 − 0.643i)14-s + (−0.284 + 0.446i)15-s + (0.231 − 0.401i)16-s − 1.35·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.28564 + 0.300655i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.28564 + 0.300655i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-64.7 - 124. i)T \) |
| good | 2 | \( 1 + (-8.85 + 15.3i)T + (-256 - 443. i)T^{2} \) |
| 5 | \( 1 + (-369. - 640. i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 7 | \( 1 + (-3.01e3 + 5.21e3i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (9.21e3 - 1.59e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + (5.87e4 + 1.01e5i)T + (-5.30e9 + 9.18e9i)T^{2} \) |
| 17 | \( 1 + 4.68e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 8.34e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + (6.60e5 + 1.14e6i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + (3.21e4 - 5.56e4i)T + (-7.25e12 - 1.25e13i)T^{2} \) |
| 31 | \( 1 + (3.70e6 + 6.42e6i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 - 3.68e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + (-1.26e7 - 2.19e7i)T + (-1.63e14 + 2.83e14i)T^{2} \) |
| 43 | \( 1 + (1.21e7 - 2.10e7i)T + (-2.51e14 - 4.35e14i)T^{2} \) |
| 47 | \( 1 + (3.51e6 - 6.09e6i)T + (-5.59e14 - 9.69e14i)T^{2} \) |
| 53 | \( 1 + 3.10e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + (-7.68e7 - 1.33e8i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-7.75e7 + 1.34e8i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-6.98e6 - 1.21e7i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + 3.45e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.01e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + (1.79e8 - 3.11e8i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-5.25e7 + 9.09e7i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 - 8.60e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + (4.74e7 - 8.21e7i)T + (-3.80e17 - 6.58e17i)T^{2} \) |
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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
−20.10397982927962636562353334532, −17.72416290349908016700487531119, −16.22875749761992780893931477108, −14.59689632598300365766963560274, −13.26001618436831057851903769356, −11.21544758644122421671496550942, −10.10643315027665598075262010907, −7.72462853471714756261785458215, −4.46148927379613480688104918487, −2.68209473141744205555985957157,
1.81237647370491557453538158365, 5.42613578544033220424454150805, 7.15143262309385683704303712431, 9.002454048319939814111549637766, 11.67772213867006817318859403127, 13.44036012816751192292088545318, 14.53094924915976436769030586205, 15.94184999666933424389305387979, 17.70928017734304094533851381247, 19.11287012126514484890560592974
