| L(s) = 1 | + 5.33·2-s + 9·3-s − 3.51·4-s + 48.0·6-s + 73.9·7-s − 189.·8-s + 81·9-s + 121·11-s − 31.6·12-s − 66.6·13-s + 394.·14-s − 899.·16-s − 808.·17-s + 432.·18-s − 900.·19-s + 665.·21-s + 645.·22-s + 3.37e3·23-s − 1.70e3·24-s − 355.·26-s + 729·27-s − 260.·28-s − 5.03e3·29-s + 2.59e3·31-s + 1.26e3·32-s + 1.08e3·33-s − 4.31e3·34-s + ⋯ |
| L(s) = 1 | + 0.943·2-s + 0.577·3-s − 0.109·4-s + 0.544·6-s + 0.570·7-s − 1.04·8-s + 0.333·9-s + 0.301·11-s − 0.0634·12-s − 0.109·13-s + 0.538·14-s − 0.877·16-s − 0.678·17-s + 0.314·18-s − 0.572·19-s + 0.329·21-s + 0.284·22-s + 1.32·23-s − 0.604·24-s − 0.103·26-s + 0.192·27-s − 0.0627·28-s − 1.11·29-s + 0.485·31-s + 0.218·32-s + 0.174·33-s − 0.640·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 - 5.33T + 32T^{2} \) |
| 7 | \( 1 - 73.9T + 1.68e4T^{2} \) |
| 13 | \( 1 + 66.6T + 3.71e5T^{2} \) |
| 17 | \( 1 + 808.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 900.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.37e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.03e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.59e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.95e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 197.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.33e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.82e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 4.85e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 6.93e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.05e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.07e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.39e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.37e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.89e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.03e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.30e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.29e4T + 8.58e9T^{2} \) |
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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
−8.923143486964621527317815011034, −8.384010970929544799253586501770, −7.21234455444380444136171686328, −6.35030521189819457458797757401, −5.21906568855720365084251337821, −4.54183567369246837171094388914, −3.66333641223519573829296696010, −2.71737279796770158005465484361, −1.54735028653630171891955200887, 0,
1.54735028653630171891955200887, 2.71737279796770158005465484361, 3.66333641223519573829296696010, 4.54183567369246837171094388914, 5.21906568855720365084251337821, 6.35030521189819457458797757401, 7.21234455444380444136171686328, 8.384010970929544799253586501770, 8.923143486964621527317815011034
