| L(s) = 1 | + 33.3·2-s − 72.5·3-s + 601.·4-s − 625·5-s − 2.42e3·6-s − 2.40e3·7-s + 2.97e3·8-s − 1.44e4·9-s − 2.08e4·10-s − 3.15e4·11-s − 4.36e4·12-s + 6.70e4·13-s − 8.01e4·14-s + 4.53e4·15-s − 2.08e5·16-s − 2.21e5·17-s − 4.80e5·18-s − 6.31e5·19-s − 3.75e5·20-s + 1.74e5·21-s − 1.05e6·22-s + 2.90e5·23-s − 2.16e5·24-s + 3.90e5·25-s + 2.23e6·26-s + 2.47e6·27-s − 1.44e6·28-s + ⋯ |
| L(s) = 1 | + 1.47·2-s − 0.517·3-s + 1.17·4-s − 0.447·5-s − 0.763·6-s − 0.377·7-s + 0.257·8-s − 0.732·9-s − 0.659·10-s − 0.650·11-s − 0.607·12-s + 0.651·13-s − 0.557·14-s + 0.231·15-s − 0.795·16-s − 0.643·17-s − 1.07·18-s − 1.11·19-s − 0.525·20-s + 0.195·21-s − 0.958·22-s + 0.216·23-s − 0.133·24-s + 0.200·25-s + 0.959·26-s + 0.896·27-s − 0.443·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 625T \) |
| 7 | \( 1 + 2.40e3T \) |
| good | 2 | \( 1 - 33.3T + 512T^{2} \) |
| 3 | \( 1 + 72.5T + 1.96e4T^{2} \) |
| 11 | \( 1 + 3.15e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 6.70e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.21e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 6.31e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.90e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.34e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.68e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 4.33e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.29e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.85e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.63e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.95e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.21e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 6.35e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.35e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 8.05e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.43e6T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.06e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.73e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.34e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.27e9T + 7.60e17T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.74341137301095514107909051078, −12.81341801636368429012462386252, −11.72884832118828843953356698347, −10.74914591814373053145302327545, −8.605145728505840786365675670757, −6.64205589479034813623097794753, −5.55900142306365682560529695878, −4.21025071740809617332742080535, −2.75944182268194186078664262244, 0,
2.75944182268194186078664262244, 4.21025071740809617332742080535, 5.55900142306365682560529695878, 6.64205589479034813623097794753, 8.605145728505840786365675670757, 10.74914591814373053145302327545, 11.72884832118828843953356698347, 12.81341801636368429012462386252, 13.74341137301095514107909051078
