| L(s) = 1 | + 13.3·2-s + 163.·3-s − 333.·4-s + 1.92e3·5-s + 2.18e3·6-s + 2.40e3·7-s − 1.12e4·8-s + 7.02e3·9-s + 2.56e4·10-s − 9.01e4·11-s − 5.44e4·12-s − 3.19e3·13-s + 3.20e4·14-s + 3.14e5·15-s + 1.98e4·16-s + 1.16e5·17-s + 9.38e4·18-s − 1.42e5·19-s − 6.41e5·20-s + 3.92e5·21-s − 1.20e6·22-s + 1.27e6·23-s − 1.84e6·24-s + 1.74e6·25-s − 4.27e4·26-s − 2.06e6·27-s − 8.00e5·28-s + ⋯ |
| L(s) = 1 | + 0.590·2-s + 1.16·3-s − 0.651·4-s + 1.37·5-s + 0.687·6-s + 0.377·7-s − 0.975·8-s + 0.356·9-s + 0.812·10-s − 1.85·11-s − 0.758·12-s − 0.0310·13-s + 0.223·14-s + 1.60·15-s + 0.0756·16-s + 0.338·17-s + 0.210·18-s − 0.250·19-s − 0.895·20-s + 0.440·21-s − 1.09·22-s + 0.949·23-s − 1.13·24-s + 0.891·25-s − 0.0183·26-s − 0.749·27-s − 0.246·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.422570531\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.422570531\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 2.40e3T \) |
| good | 2 | \( 1 - 13.3T + 512T^{2} \) |
| 3 | \( 1 - 163.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.92e3T + 1.95e6T^{2} \) |
| 11 | \( 1 + 9.01e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 3.19e3T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.16e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 1.42e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.27e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.42e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 9.67e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 8.67e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.32e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.97e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.07e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 7.07e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 6.40e6T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.69e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.16e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.44e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.60e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.89e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 8.31e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.08e6T + 3.50e17T^{2} \) |
| 97 | \( 1 + 3.15e8T + 7.60e17T^{2} \) |
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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
−20.90676915593351373585352758217, −18.75840307750738874569912665810, −17.55766236376809216049083689603, −15.06079762327990915889930007270, −13.82330808682233201677173380978, −13.11997253337513675165158030851, −9.956179202832500850256664400361, −8.424105492050637531185092440215, −5.31592821540955963675027295774, −2.67225695276816042412644794101,
2.67225695276816042412644794101, 5.31592821540955963675027295774, 8.424105492050637531185092440215, 9.956179202832500850256664400361, 13.11997253337513675165158030851, 13.82330808682233201677173380978, 15.06079762327990915889930007270, 17.55766236376809216049083689603, 18.75840307750738874569912665810, 20.90676915593351373585352758217
