| L(s) = 1 | + (13.0 − 4.76i)5-s + (−0.689 − 3.91i)7-s + (41.0 + 14.9i)11-s + (4.04 + 3.39i)13-s + (3.06 − 5.30i)17-s + (−7.42 − 12.8i)19-s + (21.3 − 121. i)23-s + (52.8 − 44.3i)25-s + (108. − 90.8i)29-s + (−46.8 + 265. i)31-s + (−27.6 − 47.8i)35-s + (100. − 173. i)37-s + (−239. − 200. i)41-s + (462. + 168. i)43-s + (10.6 + 60.1i)47-s + ⋯ |
| L(s) = 1 | + (1.17 − 0.426i)5-s + (−0.0372 − 0.211i)7-s + (1.12 + 0.409i)11-s + (0.0862 + 0.0723i)13-s + (0.0436 − 0.0756i)17-s + (−0.0896 − 0.155i)19-s + (0.193 − 1.09i)23-s + (0.422 − 0.354i)25-s + (0.692 − 0.581i)29-s + (−0.271 + 1.53i)31-s + (−0.133 − 0.231i)35-s + (0.445 − 0.771i)37-s + (−0.911 − 0.765i)41-s + (1.63 + 0.596i)43-s + (0.0329 + 0.186i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.453802410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.453802410\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-13.0 + 4.76i)T + (95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (0.689 + 3.91i)T + (-322. + 117. i)T^{2} \) |
| 11 | \( 1 + (-41.0 - 14.9i)T + (1.01e3 + 855. i)T^{2} \) |
| 13 | \( 1 + (-4.04 - 3.39i)T + (381. + 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-3.06 + 5.30i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (7.42 + 12.8i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-21.3 + 121. i)T + (-1.14e4 - 4.16e3i)T^{2} \) |
| 29 | \( 1 + (-108. + 90.8i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (46.8 - 265. i)T + (-2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (-100. + 173. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (239. + 200. i)T + (1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + (-462. - 168. i)T + (6.09e4 + 5.11e4i)T^{2} \) |
| 47 | \( 1 + (-10.6 - 60.1i)T + (-9.75e4 + 3.55e4i)T^{2} \) |
| 53 | \( 1 - 380.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-154. + 56.3i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (149. + 849. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-159. - 133. i)T + (5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-451. + 781. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-353. - 612. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (847. - 711. i)T + (8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (652. - 547. i)T + (9.92e4 - 5.63e5i)T^{2} \) |
| 89 | \( 1 + (214. + 371. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (1.16e3 + 424. i)T + (6.99e5 + 5.86e5i)T^{2} \) |
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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
−11.00831340287911607755030753273, −10.06764603663366393127879271907, −9.280047135320527169797701363349, −8.541366049516895844020258547350, −7.05234530843625196468964888453, −6.25456222962900191620148467380, −5.15552176015612952496866869877, −4.01845242308242070178362998567, −2.33834878008505984788267088381, −1.07877419663461755008576205652,
1.32369588737719280451611657918, 2.65143766712092024673389590889, 4.00732681264955167198227433722, 5.58249786267171334365554809211, 6.22029276384819579625224432227, 7.25673802740323820371550272425, 8.627922070913294708933016308264, 9.459443343390365269695254879870, 10.19653822099860653670772448260, 11.24888489329763833533960002349
