| L(s) = 1 | − 10.7·2-s − 9·3-s + 83.6·4-s + 96.7·6-s − 41.3·7-s − 555.·8-s + 81·9-s + 121·11-s − 752.·12-s + 13.6·13-s + 444.·14-s + 3.29e3·16-s − 1.14e3·17-s − 870.·18-s − 1.42e3·19-s + 372.·21-s − 1.30e3·22-s + 2.53e3·23-s + 4.99e3·24-s − 146.·26-s − 729·27-s − 3.46e3·28-s + 8.59e3·29-s + 434.·31-s − 1.76e4·32-s − 1.08e3·33-s + 1.23e4·34-s + ⋯ |
| L(s) = 1 | − 1.90·2-s − 0.577·3-s + 2.61·4-s + 1.09·6-s − 0.319·7-s − 3.06·8-s + 0.333·9-s + 0.301·11-s − 1.50·12-s + 0.0224·13-s + 0.606·14-s + 3.21·16-s − 0.960·17-s − 0.633·18-s − 0.905·19-s + 0.184·21-s − 0.573·22-s + 0.998·23-s + 1.77·24-s − 0.0425·26-s − 0.192·27-s − 0.834·28-s + 1.89·29-s + 0.0811·31-s − 3.04·32-s − 0.174·33-s + 1.82·34-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 | \( 1 + 10.7T + 32T^{2} \) |
| 7 | \( 1 + 41.3T + 1.68e4T^{2} \) |
| 13 | \( 1 - 13.6T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.14e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.42e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.53e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.59e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 434.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.44e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.23e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.69e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.69e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 9.46e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.14e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.78e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.17e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.66e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.97e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.14e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.35e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.69e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.77e4T + 8.58e9T^{2} \) |
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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
−8.991515873124622678303560472377, −8.490243565746000077361310527200, −7.39211865804201739181769543422, −6.64651948732376684061687661782, −6.13179234803344424263636725634, −4.65144343202902931375835465657, −3.06807376107113962811204007070, −1.97271925640032540788368469156, −0.916538144928096309686953261275, 0,
0.916538144928096309686953261275, 1.97271925640032540788368469156, 3.06807376107113962811204007070, 4.65144343202902931375835465657, 6.13179234803344424263636725634, 6.64651948732376684061687661782, 7.39211865804201739181769543422, 8.490243565746000077361310527200, 8.991515873124622678303560472377
