| L(s) = 1 | + 7.59·2-s − 26.8·3-s + 25.6·4-s + 25·5-s − 203.·6-s − 48.1·8-s + 477.·9-s + 189.·10-s + 698.·11-s − 688.·12-s − 469.·13-s − 670.·15-s − 1.18e3·16-s − 1.37e3·17-s + 3.62e3·18-s + 1.61e3·19-s + 641.·20-s + 5.30e3·22-s − 467.·23-s + 1.29e3·24-s + 625·25-s − 3.56e3·26-s − 6.28e3·27-s + 2.61e3·29-s − 5.09e3·30-s − 5.57e3·31-s − 7.46e3·32-s + ⋯ |
| L(s) = 1 | + 1.34·2-s − 1.72·3-s + 0.801·4-s + 0.447·5-s − 2.31·6-s − 0.266·8-s + 1.96·9-s + 0.600·10-s + 1.73·11-s − 1.38·12-s − 0.769·13-s − 0.769·15-s − 1.15·16-s − 1.15·17-s + 2.63·18-s + 1.02·19-s + 0.358·20-s + 2.33·22-s − 0.184·23-s + 0.458·24-s + 0.200·25-s − 1.03·26-s − 1.65·27-s + 0.577·29-s − 1.03·30-s − 1.04·31-s − 1.28·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - 25T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 7.59T + 32T^{2} \) |
| 3 | \( 1 + 26.8T + 243T^{2} \) |
| 11 | \( 1 - 698.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 469.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.37e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.61e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 467.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.61e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.57e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.52e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.37e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.99e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.99e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.33e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.38e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.33e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 7.25e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.71e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.65e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.47e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.58e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.18e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.65e4T + 8.58e9T^{2} \) |
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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
−11.31432802777673487047154780685, −10.06902675291449417732224397882, −9.063196812950150656865388753247, −6.80063578983768814913908050677, −6.55370047502746144273819266650, −5.36351313732453172264208167768, −4.78214155804686530220317178161, −3.63249359873838776552880631771, −1.65053916787079488975885624541, 0,
1.65053916787079488975885624541, 3.63249359873838776552880631771, 4.78214155804686530220317178161, 5.36351313732453172264208167768, 6.55370047502746144273819266650, 6.80063578983768814913908050677, 9.063196812950150656865388753247, 10.06902675291449417732224397882, 11.31432802777673487047154780685
