| L(s) = 1 | + 2.81·2-s + 6.54·3-s − 24.0·4-s − 45.9·5-s + 18.4·6-s − 157.·8-s − 200.·9-s − 129.·10-s − 551.·11-s − 157.·12-s + 1.09e3·13-s − 301.·15-s + 326.·16-s − 1.18e3·17-s − 563.·18-s − 1.16e3·19-s + 1.10e3·20-s − 1.55e3·22-s + 44.3·23-s − 1.03e3·24-s − 1.00e3·25-s + 3.07e3·26-s − 2.90e3·27-s + 3.32e3·29-s − 847.·30-s + 8.78e3·31-s + 5.96e3·32-s + ⋯ |
| L(s) = 1 | + 0.497·2-s + 0.420·3-s − 0.752·4-s − 0.822·5-s + 0.209·6-s − 0.872·8-s − 0.823·9-s − 0.409·10-s − 1.37·11-s − 0.316·12-s + 1.79·13-s − 0.345·15-s + 0.318·16-s − 0.990·17-s − 0.409·18-s − 0.741·19-s + 0.618·20-s − 0.684·22-s + 0.0174·23-s − 0.366·24-s − 0.323·25-s + 0.893·26-s − 0.765·27-s + 0.735·29-s − 0.171·30-s + 1.64·31-s + 1.03·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| good | 2 | \( 1 - 2.81T + 32T^{2} \) |
| 3 | \( 1 - 6.54T + 243T^{2} \) |
| 5 | \( 1 + 45.9T + 3.12e3T^{2} \) |
| 11 | \( 1 + 551.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.09e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.18e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.16e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 44.3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.32e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.78e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.55e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.27e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 96.7T + 1.47e8T^{2} \) |
| 47 | \( 1 - 7.67e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.19e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 9.85e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.85e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.75e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.13e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.85e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.52T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.50e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.36e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.11e3T + 8.58e9T^{2} \) |
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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
−13.76339493296830452763038246471, −13.24634674106337374779345853472, −11.81213418936736043791719041296, −10.58241922657192764114216786963, −8.717782316119673337279916098100, −8.183056734645235509753885554998, −6.03373819024320781084429844665, −4.44770378485442115297806229410, −3.09666836325306029808568338199, 0,
3.09666836325306029808568338199, 4.44770378485442115297806229410, 6.03373819024320781084429844665, 8.183056734645235509753885554998, 8.717782316119673337279916098100, 10.58241922657192764114216786963, 11.81213418936736043791719041296, 13.24634674106337374779345853472, 13.76339493296830452763038246471
