| L(s) = 1 | − 4.24·5-s − 61.7·7-s − 87.2·11-s + 238i·13-s − 304. i·17-s + 617. i·19-s + 610. i·23-s − 607·25-s − 120.·29-s + 1.54e3·31-s + 261.·35-s − 728i·37-s + 1.64e3i·41-s + 246. i·43-s + 1.48e3i·47-s + ⋯ |
| L(s) = 1 | − 0.169·5-s − 1.25·7-s − 0.721·11-s + 1.40i·13-s − 1.05i·17-s + 1.70i·19-s + 1.15i·23-s − 0.971·25-s − 0.142·29-s + 1.60·31-s + 0.213·35-s − 0.531i·37-s + 0.976i·41-s + 0.133i·43-s + 0.671i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.2376691548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2376691548\) |
| \(L(3)\) |
|
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 | \( 1 + 4.24T + 625T^{2} \) |
| 7 | \( 1 + 61.7T + 2.40e3T^{2} \) |
| 11 | \( 1 + 87.2T + 1.46e4T^{2} \) |
| 13 | \( 1 - 238iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 304. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 617. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 610. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 120.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.54e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 728iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 1.64e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 246. iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 1.48e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.74e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 6.28e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 2.23e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 2.59e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 6.37e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 2.49e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 4.75e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 1.19e4T + 4.74e7T^{2} \) |
| 89 | \( 1 + 1.15e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.40e4T + 8.85e7T^{2} \) |
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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
−9.333137355735041055549442027501, −8.070765123436712207154744704641, −7.40434768104014560348122656998, −6.43578808781166357231518356981, −5.80691734288866254641651706787, −4.61612904519355078481493602128, −3.67058825311190315353027716827, −2.79628888143239760271934178577, −1.57618188575455776546990363235, −0.06995403506834462841484555978,
0.69775511750014774992085612834, 2.49417774513217001864898773628, 3.12323838242308619594324697640, 4.22114390159217020777415210156, 5.30195318710543252283688361329, 6.17510658855306456569914498563, 6.89588462481496215435466432461, 7.923085468326604890809864297169, 8.571101176013551210358078025305, 9.559166094031159746211440034469
