| L(s) = 1 | + 5.60·2-s + 5.22·3-s − 480.·4-s − 529.·5-s + 29.2·6-s − 3.70e3·7-s − 5.56e3·8-s − 1.96e4·9-s − 2.97e3·10-s − 1.46e4·11-s − 2.51e3·12-s + 3.09e4·13-s − 2.07e4·14-s − 2.76e3·15-s + 2.14e5·16-s + 2.50e5·17-s − 1.10e5·18-s + 4.38e5·19-s + 2.54e5·20-s − 1.93e4·21-s − 8.21e4·22-s − 1.78e6·23-s − 2.90e4·24-s − 1.67e6·25-s + 1.73e5·26-s − 2.05e5·27-s + 1.78e6·28-s + ⋯ |
| L(s) = 1 | + 0.247·2-s + 0.0372·3-s − 0.938·4-s − 0.379·5-s + 0.00922·6-s − 0.583·7-s − 0.480·8-s − 0.998·9-s − 0.0939·10-s − 0.301·11-s − 0.0349·12-s + 0.300·13-s − 0.144·14-s − 0.0141·15-s + 0.819·16-s + 0.728·17-s − 0.247·18-s + 0.771·19-s + 0.355·20-s − 0.0217·21-s − 0.0747·22-s − 1.32·23-s − 0.0178·24-s − 0.856·25-s + 0.0745·26-s − 0.0744·27-s + 0.547·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + 1.46e4T \) |
| good | 2 | \( 1 - 5.60T + 512T^{2} \) |
| 3 | \( 1 - 5.22T + 1.96e4T^{2} \) |
| 5 | \( 1 + 529.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 3.70e3T + 4.03e7T^{2} \) |
| 13 | \( 1 - 3.09e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.50e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 4.38e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.78e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.59e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 2.09e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 2.62e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.91e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.01e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.45e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.32e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.40e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.32e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 4.13e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.81e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 4.13e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.90e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.29e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 7.73e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.00e9T + 7.60e17T^{2} \) |
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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
−17.74306310543556674993185862740, −16.22589999015850201711488560530, −14.55570180570617671039107538143, −13.35261627139822178813434659711, −11.83467452835342509128651440808, −9.746389424675796263351221684348, −8.158399499325111867061010824084, −5.63559385554338713809422228282, −3.53286253696665252586690407602, 0,
3.53286253696665252586690407602, 5.63559385554338713809422228282, 8.158399499325111867061010824084, 9.746389424675796263351221684348, 11.83467452835342509128651440808, 13.35261627139822178813434659711, 14.55570180570617671039107538143, 16.22589999015850201711488560530, 17.74306310543556674993185862740
