| L(s) = 1 | + (3.23 − 2.35i)2-s + (2.78 − 8.55i)3-s + (4.94 − 15.2i)4-s + (−10.7 + 54.8i)5-s + (−11.1 − 34.2i)6-s + 36.7·7-s + (−19.7 − 60.8i)8-s + (−65.5 − 47.6i)9-s + (94.1 + 202. i)10-s + (−582. + 423. i)11-s + (−116. − 84.6i)12-s + (−651. − 473. i)13-s + (118. − 86.4i)14-s + (439. + 244. i)15-s + (−207. − 150. i)16-s + (79.4 + 244. i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (0.178 − 0.549i)3-s + (0.154 − 0.475i)4-s + (−0.192 + 0.981i)5-s + (−0.126 − 0.388i)6-s + 0.283·7-s + (−0.109 − 0.336i)8-s + (−0.269 − 0.195i)9-s + (0.297 + 0.641i)10-s + (−1.45 + 1.05i)11-s + (−0.233 − 0.169i)12-s + (−1.06 − 0.776i)13-s + (0.162 − 0.117i)14-s + (0.504 + 0.280i)15-s + (−0.202 − 0.146i)16-s + (0.0666 + 0.205i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 - 0.777i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.2768586298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2768586298\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + (-2.78 + 8.55i)T \) |
| 5 | \( 1 + (10.7 - 54.8i)T \) |
| good | 7 | \( 1 - 36.7T + 1.68e4T^{2} \) |
| 11 | \( 1 + (582. - 423. i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (651. + 473. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-79.4 - 244. i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (416. + 1.28e3i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (3.60e3 - 2.61e3i)T + (1.98e6 - 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-148. + 455. i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-476. - 1.46e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-5.60e3 - 4.07e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-1.96e3 - 1.42e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 - 1.47e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.26e3 - 3.88e3i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (239. - 736. i)T + (-3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (3.89e3 + 2.82e3i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (1.63e4 - 1.18e4i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (-6.42e3 - 1.97e4i)T + (-1.09e9 + 7.93e8i)T^{2} \) |
| 71 | \( 1 + (-8.48e3 + 2.61e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (5.43e4 - 3.95e4i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-2.79e4 + 8.60e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (2.45e4 + 7.54e4i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + (-2.63e4 + 1.91e4i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (7.40e3 - 2.27e4i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−12.53767166805684483422808146205, −11.59694550692194110759522731042, −10.51046003109856188621870235482, −9.798586571797944529738937127489, −7.85287718974311358825329564001, −7.31225208575887164048088973508, −5.89026854089362993218101790754, −4.62883467326959469670222914404, −2.95483284707906455619254783760, −2.11802308656238251449077921789,
0.06709525062457607122824755203, 2.41189046938453887643305257248, 4.05221817908080880593900691352, 4.98104930123082914541350415168, 5.94062936536014001523936623099, 7.75759846986771268320266815238, 8.363564985099467956128786456827, 9.596295097594329708374952211652, 10.79541378117830102937989865399, 11.96945128069235879706764760222
