| L(s) = 1 | + 15.2·5-s + 55.6·7-s − 98.3·11-s + 135. i·13-s − 285. i·17-s + 161. i·19-s − 48.6i·23-s − 392.·25-s + 915.·29-s + 877.·31-s + 848.·35-s + 2.34e3i·37-s + 361. i·41-s − 683. i·43-s + 4.00e3i·47-s + ⋯ |
| L(s) = 1 | + 0.609·5-s + 1.13·7-s − 0.813·11-s + 0.804i·13-s − 0.987i·17-s + 0.448i·19-s − 0.0920i·23-s − 0.628·25-s + 1.08·29-s + 0.913·31-s + 0.692·35-s + 1.71i·37-s + 0.215i·41-s − 0.369i·43-s + 1.81i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.605600332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.605600332\) |
| \(L(3)\) |
|
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 | \( 1 - 15.2T + 625T^{2} \) |
| 7 | \( 1 - 55.6T + 2.40e3T^{2} \) |
| 11 | \( 1 + 98.3T + 1.46e4T^{2} \) |
| 13 | \( 1 - 135. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 285. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 161. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 48.6iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 915.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 877.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 2.34e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 361. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 683. iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 4.00e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.27e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 188.T + 1.21e7T^{2} \) |
| 61 | \( 1 + 1.61e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 4.90e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 2.41e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 3.70e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 6.11e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 1.29e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 299. iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 8.29e3T + 8.85e7T^{2} \) |
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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
−9.484890785748215555954069027443, −8.402005911902945573045479488094, −7.889818586370668353953399982299, −6.86681862112960409112838925264, −5.96789710531389752106014331019, −4.97962647884382990776855505746, −4.46146261075754486914995571813, −2.94193012424312029936966760069, −2.02819125829412140568164547309, −1.03211805301719522040145525123,
0.57417509373324222336163619292, 1.78996318476002021244077287157, 2.61757460752692082287367006607, 3.92567175074579172636086326958, 5.01779413929069298528667606779, 5.57609403632020481091573601875, 6.53887746069801963090192578432, 7.72112772466675835071261545171, 8.182449885777606611229546825390, 9.031445557912791969947904868326
