| L(s) = 1 | − 250.·3-s + 1.55e3·5-s + 8.32e3·7-s + 4.30e4·9-s − 3.07e4·11-s + 2.85e4·13-s − 3.89e5·15-s − 6.37e5·17-s − 1.05e5·19-s − 2.08e6·21-s + 5.11e5·23-s + 4.66e5·25-s − 5.85e6·27-s + 7.81e5·29-s + 2.83e6·31-s + 7.70e6·33-s + 1.29e7·35-s + 1.22e7·37-s − 7.15e6·39-s − 6.83e6·41-s − 3.84e7·43-s + 6.69e7·45-s + 1.30e7·47-s + 2.90e7·49-s + 1.59e8·51-s − 2.42e7·53-s − 4.78e7·55-s + ⋯ |
| L(s) = 1 | − 1.78·3-s + 1.11·5-s + 1.31·7-s + 2.18·9-s − 0.633·11-s + 0.277·13-s − 1.98·15-s − 1.85·17-s − 0.186·19-s − 2.34·21-s + 0.380·23-s + 0.238·25-s − 2.12·27-s + 0.205·29-s + 0.550·31-s + 1.13·33-s + 1.45·35-s + 1.07·37-s − 0.495·39-s − 0.377·41-s − 1.71·43-s + 2.43·45-s + 0.389·47-s + 0.719·49-s + 3.30·51-s − 0.422·53-s − 0.704·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 13 | \( 1 - 2.85e4T \) |
| good | 3 | \( 1 + 250.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.55e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 8.32e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.07e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 6.37e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 1.05e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 5.11e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 7.81e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.83e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.22e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 6.83e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.84e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.30e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.42e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.63e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.90e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 7.22e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.65e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.42e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.64e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.46e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 3.65e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.98e7T + 7.60e17T^{2} \) |
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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
−10.66704370040835682598586082630, −9.551777206714869085647467455149, −8.201451613993912249269769036199, −6.80910038270533237670810923221, −6.05247961861548853021674216887, −5.08359343174844389881588039583, −4.53054225710761360075774885240, −2.16573680279845600868060409414, −1.26658690714945075218574020483, 0,
1.26658690714945075218574020483, 2.16573680279845600868060409414, 4.53054225710761360075774885240, 5.08359343174844389881588039583, 6.05247961861548853021674216887, 6.80910038270533237670810923221, 8.201451613993912249269769036199, 9.551777206714869085647467455149, 10.66704370040835682598586082630
