| L(s) = 1 | + 10.3·2-s + 10.7·3-s + 74.7·4-s + 25·5-s + 111.·6-s + 442.·8-s − 126.·9-s + 258.·10-s − 262.·11-s + 806.·12-s + 688.·13-s + 269.·15-s + 2.17e3·16-s + 1.67e3·17-s − 1.31e3·18-s + 925.·19-s + 1.86e3·20-s − 2.71e3·22-s + 4.60e3·23-s + 4.76e3·24-s + 625·25-s + 7.11e3·26-s − 3.98e3·27-s − 3.41e3·29-s + 2.78e3·30-s − 5.53e3·31-s + 8.34e3·32-s + ⋯ |
| L(s) = 1 | + 1.82·2-s + 0.691·3-s + 2.33·4-s + 0.447·5-s + 1.26·6-s + 2.44·8-s − 0.521·9-s + 0.816·10-s − 0.655·11-s + 1.61·12-s + 1.13·13-s + 0.309·15-s + 2.12·16-s + 1.40·17-s − 0.953·18-s + 0.587·19-s + 1.04·20-s − 1.19·22-s + 1.81·23-s + 1.68·24-s + 0.200·25-s + 2.06·26-s − 1.05·27-s − 0.753·29-s + 0.564·30-s − 1.03·31-s + 1.44·32-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
| \(L(3)\) |
\(\approx\) |
\(8.709659772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.709659772\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 10.3T + 32T^{2} \) |
| 3 | \( 1 - 10.7T + 243T^{2} \) |
| 11 | \( 1 + 262.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 688.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.67e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 925.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.60e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.41e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.53e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.49e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.49e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.96e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.12e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.13e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.47e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.23e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.49e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.09e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 9.20e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.35e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.55e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.28e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.64e5T + 8.58e9T^{2} \) |
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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
−11.40674277036700236007737711225, −10.71580697532801239594977059363, −9.300212069311730530405775479494, −8.034988356819326129990451482777, −6.94910107441065202534613987658, −5.69822328825018331589130398597, −5.17153433562635558031467742947, −3.50543527685117736291728843770, −3.03261954702476103570770879756, −1.62815860822366776638144701547,
1.62815860822366776638144701547, 3.03261954702476103570770879756, 3.50543527685117736291728843770, 5.17153433562635558031467742947, 5.69822328825018331589130398597, 6.94910107441065202534613987658, 8.034988356819326129990451482777, 9.300212069311730530405775479494, 10.71580697532801239594977059363, 11.40674277036700236007737711225
