L(s) = 1 | − 0.703·2-s − 141.·3-s − 511.·4-s − 2.24e3·5-s + 99.6·6-s + 2.40e3·7-s + 720.·8-s + 395.·9-s + 1.57e3·10-s + 7.33e4·11-s + 7.24e4·12-s + 2.85e4·13-s − 1.68e3·14-s + 3.18e5·15-s + 2.61e5·16-s − 4.55e5·17-s − 278.·18-s + 1.01e6·19-s + 1.14e6·20-s − 3.40e5·21-s − 5.15e4·22-s − 1.84e6·23-s − 1.02e5·24-s + 3.08e6·25-s − 2.00e4·26-s + 2.73e6·27-s − 1.22e6·28-s + ⋯ |
L(s) = 1 | − 0.0310·2-s − 1.01·3-s − 0.999·4-s − 1.60·5-s + 0.0314·6-s + 0.377·7-s + 0.0621·8-s + 0.0201·9-s + 0.0499·10-s + 1.50·11-s + 1.00·12-s + 0.277·13-s − 0.0117·14-s + 1.62·15-s + 0.997·16-s − 1.32·17-s − 0.000625·18-s + 1.79·19-s + 1.60·20-s − 0.381·21-s − 0.0469·22-s − 1.37·23-s − 0.0627·24-s + 1.57·25-s − 0.00862·26-s + 0.989·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - 2.40e3T \) |
| 13 | \( 1 - 2.85e4T \) |
good | 2 | \( 1 + 0.703T + 512T^{2} \) |
| 3 | \( 1 + 141.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 2.24e3T + 1.95e6T^{2} \) |
| 11 | \( 1 - 7.33e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 4.55e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.01e6T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.84e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.40e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 7.24e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.23e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 4.18e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.21e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.54e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 9.65e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 9.86e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.20e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.79e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.78e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.01e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.14e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.27e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 7.30e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 8.70e7T + 7.60e17T^{2} \) |
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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
−11.63982466199503618450682240314, −11.09225872815617637067357770059, −9.392246756927966413080897805819, −8.387156619584199865404786731639, −7.22239752201170860340466064748, −5.74620119906741399013493481912, −4.43800222899875627342762895729, −3.73262071568285528940593985177, −0.979330092345369840804590889079, 0,
0.979330092345369840804590889079, 3.73262071568285528940593985177, 4.43800222899875627342762895729, 5.74620119906741399013493481912, 7.22239752201170860340466064748, 8.387156619584199865404786731639, 9.392246756927966413080897805819, 11.09225872815617637067357770059, 11.63982466199503618450682240314