L(s) = 1 | + 16·2-s + 270.·3-s + 256·4-s + 1.36e3·5-s + 4.33e3·6-s + 4.09e3·8-s + 5.36e4·9-s + 2.19e4·10-s − 1.09e4·11-s + 6.93e4·12-s − 7.96e3·13-s + 3.70e5·15-s + 6.55e4·16-s − 3.24e5·17-s + 8.58e5·18-s − 2.31e5·19-s + 3.50e5·20-s − 1.74e5·22-s + 2.10e6·23-s + 1.10e6·24-s − 7.74e4·25-s − 1.27e5·26-s + 9.20e6·27-s − 5.46e6·29-s + 5.93e6·30-s − 1.97e6·31-s + 1.04e6·32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.93·3-s + 0.5·4-s + 0.979·5-s + 1.36·6-s + 0.353·8-s + 2.72·9-s + 0.692·10-s − 0.224·11-s + 0.965·12-s − 0.0773·13-s + 1.89·15-s + 0.250·16-s − 0.940·17-s + 1.92·18-s − 0.408·19-s + 0.489·20-s − 0.159·22-s + 1.56·23-s + 0.682·24-s − 0.0396·25-s − 0.0547·26-s + 3.33·27-s − 1.43·29-s + 1.33·30-s − 0.384·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(7.883762343\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.883762343\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 16T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 270.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.36e3T + 1.95e6T^{2} \) |
| 11 | \( 1 + 1.09e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 7.96e3T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.24e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.31e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.10e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.46e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.97e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 4.76e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 5.57e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.11e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.31e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 9.02e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 7.29e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 7.43e6T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.05e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.10e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.04e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 8.87e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.41e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.48e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.16e9T + 7.60e17T^{2} \) |
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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
−12.87562364441926446746276109928, −10.90854856062389466044211373559, −9.668350605724094584073888359767, −8.932055737598557278999216396641, −7.68667682913807506803841566493, −6.57382117571000237553778493140, −4.88139256413102479736549949455, −3.57048927424647125138485048197, −2.47651340214972685233681464009, −1.67434932661066449156776031991,
1.67434932661066449156776031991, 2.47651340214972685233681464009, 3.57048927424647125138485048197, 4.88139256413102479736549949455, 6.57382117571000237553778493140, 7.68667682913807506803841566493, 8.932055737598557278999216396641, 9.668350605724094584073888359767, 10.90854856062389466044211373559, 12.87562364441926446746276109928