| L(s) = 1 | − 10.0·2-s − 9·3-s + 68.9·4-s + 90.4·6-s + 29.0·7-s − 370.·8-s + 81·9-s + 121·11-s − 620.·12-s − 1.02e3·13-s − 291.·14-s + 1.52e3·16-s + 1.50e3·17-s − 813.·18-s + 1.64e3·19-s − 261.·21-s − 1.21e3·22-s − 1.47e3·23-s + 3.33e3·24-s + 1.02e4·26-s − 729·27-s + 2.00e3·28-s − 4.57e3·29-s + 7.53e3·31-s − 3.40e3·32-s − 1.08e3·33-s − 1.51e4·34-s + ⋯ |
| L(s) = 1 | − 1.77·2-s − 0.577·3-s + 2.15·4-s + 1.02·6-s + 0.224·7-s − 2.04·8-s + 0.333·9-s + 0.301·11-s − 1.24·12-s − 1.67·13-s − 0.397·14-s + 1.48·16-s + 1.26·17-s − 0.591·18-s + 1.04·19-s − 0.129·21-s − 0.535·22-s − 0.582·23-s + 1.18·24-s + 2.98·26-s − 0.192·27-s + 0.482·28-s − 1.00·29-s + 1.40·31-s − 0.588·32-s − 0.174·33-s − 2.25·34-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 | \( 1 + 10.0T + 32T^{2} \) |
| 7 | \( 1 - 29.0T + 1.68e4T^{2} \) |
| 13 | \( 1 + 1.02e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.50e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.64e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.47e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.57e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.53e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.40e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.62e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.28e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 5.98e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.84e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.59e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.13e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.91e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.75e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.86e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.27e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.13e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.83e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.45e4T + 8.58e9T^{2} \) |
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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
−9.368062234102189195438552596940, −8.050742227031890438222820570905, −7.60735179604719338816571931994, −6.82436348068163103312596629269, −5.78965211775070673157921535766, −4.76957189283136033481948035707, −3.12245203968317520079696524477, −1.93389099870098101383101916392, −0.970374087862571651213944914098, 0,
0.970374087862571651213944914098, 1.93389099870098101383101916392, 3.12245203968317520079696524477, 4.76957189283136033481948035707, 5.78965211775070673157921535766, 6.82436348068163103312596629269, 7.60735179604719338816571931994, 8.050742227031890438222820570905, 9.368062234102189195438552596940
