| L(s) = 1 | + 1.21·5-s − 24.7i·7-s − 108. i·11-s + 55.5·13-s + 463.·17-s + 406. i·19-s + 122. i·23-s − 623.·25-s + 101.·29-s − 1.34e3i·31-s − 30.1i·35-s − 2.65e3·37-s + 1.66e3·41-s − 181. i·43-s − 2.71e3i·47-s + ⋯ |
| L(s) = 1 | + 0.0486·5-s − 0.505i·7-s − 0.899i·11-s + 0.328·13-s + 1.60·17-s + 1.12i·19-s + 0.232i·23-s − 0.997·25-s + 0.120·29-s − 1.39i·31-s − 0.0245i·35-s − 1.94·37-s + 0.991·41-s − 0.0982i·43-s − 1.22i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.771426036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.771426036\) |
| \(L(3)\) |
|
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 | \( 1 - 1.21T + 625T^{2} \) |
| 7 | \( 1 + 24.7iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 108. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 55.5T + 2.85e4T^{2} \) |
| 17 | \( 1 - 463.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 406. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 122. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 101.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.34e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 2.65e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 1.66e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 181. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 2.71e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 2.17e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 2.14e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 5.23e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 2.81e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 8.04e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 4.17e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 9.14e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 1.02e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 2.30e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 2.29e3T + 8.85e7T^{2} \) |
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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
−10.07731082783855730103518448575, −9.025950704329039768557166374803, −8.020532112937324728558985570901, −7.41086507166706237348977553130, −6.05534094963163961055292216253, −5.51004459851188281834858734581, −3.99496041110997517617752882185, −3.29275622202748492389294255875, −1.68937674848484834684362009722, −0.48215990140188257288387630593,
1.19066936311643474382510110944, 2.45676899899810270783051474854, 3.61358053809685049539002720702, 4.87004131526064915889825322818, 5.68275888316339806602421460557, 6.80794237551553730104672105035, 7.64861186894341426552151242364, 8.647808190551609992805273026021, 9.492014479543106375391187004884, 10.27052320160886257953226869227
