L(s) = 1 | + 49.3·2-s − 682.·3-s − 5.75e3·4-s − 1.56e4·5-s − 3.36e4·6-s − 1.61e5·7-s − 6.88e5·8-s − 1.12e6·9-s − 7.71e5·10-s + 5.82e6·11-s + 3.92e6·12-s + 1.51e7·13-s − 7.99e6·14-s + 1.06e7·15-s + 1.31e7·16-s − 9.09e7·17-s − 5.57e7·18-s − 1.47e8·19-s + 8.99e7·20-s + 1.10e8·21-s + 2.87e8·22-s − 5.39e8·23-s + 4.69e8·24-s + 2.44e8·25-s + 7.48e8·26-s + 1.85e9·27-s + 9.32e8·28-s + ⋯ |
L(s) = 1 | + 0.545·2-s − 0.540·3-s − 0.702·4-s − 0.447·5-s − 0.294·6-s − 0.520·7-s − 0.928·8-s − 0.708·9-s − 0.243·10-s + 0.991·11-s + 0.379·12-s + 0.871·13-s − 0.283·14-s + 0.241·15-s + 0.196·16-s − 0.913·17-s − 0.386·18-s − 0.720·19-s + 0.314·20-s + 0.281·21-s + 0.540·22-s − 0.760·23-s + 0.501·24-s + 0.199·25-s + 0.475·26-s + 0.922·27-s + 0.365·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(7)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{15}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + 1.56e4T \) |
good | 2 | \( 1 - 49.3T + 8.19e3T^{2} \) |
| 3 | \( 1 + 682.T + 1.59e6T^{2} \) |
| 7 | \( 1 + 1.61e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 5.82e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 1.51e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + 9.09e7T + 9.90e15T^{2} \) |
| 19 | \( 1 + 1.47e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 5.39e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 5.51e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 7.83e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 2.87e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 3.61e9T + 9.25e20T^{2} \) |
| 43 | \( 1 + 3.01e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 1.86e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 2.06e11T + 2.60e22T^{2} \) |
| 59 | \( 1 - 3.80e11T + 1.04e23T^{2} \) |
| 61 | \( 1 + 3.75e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 1.19e12T + 5.48e23T^{2} \) |
| 71 | \( 1 - 6.59e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 3.87e11T + 1.67e24T^{2} \) |
| 79 | \( 1 + 3.71e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 7.97e11T + 8.87e24T^{2} \) |
| 89 | \( 1 + 2.74e11T + 2.19e25T^{2} \) |
| 97 | \( 1 - 1.19e13T + 6.73e25T^{2} \) |
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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
−19.94221346413179917034256244813, −18.23499171556847430907301785184, −16.67736664763878180986342979023, −14.73718640817609947753306396156, −13.06090388683822756072475660153, −11.42476079684535018007315698505, −8.941977158864779004370207755072, −6.04262873600789091902022133204, −3.91275303115995055431909322721, 0,
3.91275303115995055431909322721, 6.04262873600789091902022133204, 8.941977158864779004370207755072, 11.42476079684535018007315698505, 13.06090388683822756072475660153, 14.73718640817609947753306396156, 16.67736664763878180986342979023, 18.23499171556847430907301785184, 19.94221346413179917034256244813