| L(s) = 1 | + 0.134·2-s − 31.9·4-s − 25·5-s − 118.·7-s − 8.63·8-s − 3.37·10-s + 121·11-s + 1.11e3·13-s − 15.9·14-s + 1.02e3·16-s − 1.76e3·17-s − 2.12e3·19-s + 799.·20-s + 16.3·22-s + 3.65e3·23-s + 625·25-s + 150.·26-s + 3.77e3·28-s + 2.91e3·29-s + 9.02e3·31-s + 414.·32-s − 237.·34-s + 2.95e3·35-s − 800.·37-s − 286.·38-s + 215.·40-s + 6.49e3·41-s + ⋯ |
| L(s) = 1 | + 0.0238·2-s − 0.999·4-s − 0.447·5-s − 0.911·7-s − 0.0477·8-s − 0.0106·10-s + 0.301·11-s + 1.83·13-s − 0.0217·14-s + 0.998·16-s − 1.47·17-s − 1.34·19-s + 0.446·20-s + 0.00719·22-s + 1.44·23-s + 0.200·25-s + 0.0436·26-s + 0.910·28-s + 0.643·29-s + 1.68·31-s + 0.0715·32-s − 0.0352·34-s + 0.407·35-s − 0.0960·37-s − 0.0321·38-s + 0.0213·40-s + 0.602·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 - 0.134T + 32T^{2} \) |
| 7 | \( 1 + 118.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 1.11e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.76e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.12e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.65e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.91e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.02e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 800.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 6.49e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.26e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.69e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.99e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.11e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.10e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.97e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.82e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.59e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.47e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.41e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.31e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.00e4T + 8.58e9T^{2} \) |
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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
−9.533012647258615300730631652072, −8.721503301796593680724576926204, −8.297522550840383249393961879463, −6.68558298942222680191872363263, −6.17907410125700228345446363909, −4.67017596666384814062072328556, −3.97896975036962744414263244328, −2.95879034450215078563120476921, −1.10315431062665439475155715055, 0,
1.10315431062665439475155715055, 2.95879034450215078563120476921, 3.97896975036962744414263244328, 4.67017596666384814062072328556, 6.17907410125700228345446363909, 6.68558298942222680191872363263, 8.297522550840383249393961879463, 8.721503301796593680724576926204, 9.533012647258615300730631652072
