| L(s) = 1 | − 85.4·2-s − 243·3-s + 5.25e3·4-s + 8.05e3·5-s + 2.07e4·6-s + 1.68e4·7-s − 2.73e5·8-s + 5.90e4·9-s − 6.88e5·10-s − 7.63e5·11-s − 1.27e6·12-s − 3.71e5·13-s − 1.43e6·14-s − 1.95e6·15-s + 1.26e7·16-s − 1.13e7·17-s − 5.04e6·18-s + 1.19e7·19-s + 4.22e7·20-s − 4.08e6·21-s + 6.52e7·22-s − 2.31e7·23-s + 6.65e7·24-s + 1.60e7·25-s + 3.17e7·26-s − 1.43e7·27-s + 8.82e7·28-s + ⋯ |
| L(s) = 1 | − 1.88·2-s − 0.577·3-s + 2.56·4-s + 1.15·5-s + 1.09·6-s + 0.377·7-s − 2.95·8-s + 0.333·9-s − 2.17·10-s − 1.43·11-s − 1.48·12-s − 0.277·13-s − 0.713·14-s − 0.665·15-s + 3.01·16-s − 1.93·17-s − 0.629·18-s + 1.10·19-s + 2.95·20-s − 0.218·21-s + 2.69·22-s − 0.749·23-s + 1.70·24-s + 0.328·25-s + 0.523·26-s − 0.192·27-s + 0.969·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 243T \) |
| 7 | \( 1 - 1.68e4T \) |
| 13 | \( 1 + 3.71e5T \) |
| good | 2 | \( 1 + 85.4T + 2.04e3T^{2} \) |
| 5 | \( 1 - 8.05e3T + 4.88e7T^{2} \) |
| 11 | \( 1 + 7.63e5T + 2.85e11T^{2} \) |
| 17 | \( 1 + 1.13e7T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.19e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 2.31e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 6.17e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 2.82e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 1.13e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 8.17e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.03e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 9.12e8T + 2.47e18T^{2} \) |
| 53 | \( 1 - 3.83e8T + 9.26e18T^{2} \) |
| 59 | \( 1 - 7.57e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 4.32e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 5.01e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 2.74e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.00e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 1.20e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 5.80e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 2.43e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.17e11T + 7.15e21T^{2} \) |
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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
−9.777287630671043035344966911834, −8.628404250360211464623166092440, −7.82137958389638331355639666131, −6.78339892411329792073388619629, −5.99370302629077978797388497127, −4.91409713316177915303908986263, −2.60407710087956823995740659731, −2.06582430196036901970112772643, −0.945342638312064250656540050646, 0,
0.945342638312064250656540050646, 2.06582430196036901970112772643, 2.60407710087956823995740659731, 4.91409713316177915303908986263, 5.99370302629077978797388497127, 6.78339892411329792073388619629, 7.82137958389638331355639666131, 8.628404250360211464623166092440, 9.777287630671043035344966911834
