| L(s) = 1 | + 9.37·3-s − 462.·5-s + 343·7-s − 2.09e3·9-s − 3.88e3·11-s + 1.15e4·13-s − 4.33e3·15-s − 1.52e4·17-s + 3.24e4·19-s + 3.21e3·21-s + 5.61e4·23-s + 1.35e5·25-s − 4.01e4·27-s + 2.64e4·29-s − 4.54e4·31-s − 3.63e4·33-s − 1.58e5·35-s + 5.55e5·37-s + 1.08e5·39-s + 3.06e5·41-s − 7.80e5·43-s + 9.70e5·45-s − 5.31e5·47-s + 1.17e5·49-s − 1.42e5·51-s + 3.63e5·53-s + 1.79e6·55-s + ⋯ |
| L(s) = 1 | + 0.200·3-s − 1.65·5-s + 0.377·7-s − 0.959·9-s − 0.879·11-s + 1.46·13-s − 0.331·15-s − 0.752·17-s + 1.08·19-s + 0.0757·21-s + 0.962·23-s + 1.73·25-s − 0.392·27-s + 0.201·29-s − 0.274·31-s − 0.176·33-s − 0.625·35-s + 1.80·37-s + 0.293·39-s + 0.694·41-s − 1.49·43-s + 1.58·45-s − 0.746·47-s + 0.142·49-s − 0.150·51-s + 0.335·53-s + 1.45·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 343T \) |
| good | 3 | \( 1 - 9.37T + 2.18e3T^{2} \) |
| 5 | \( 1 + 462.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 3.88e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.15e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.52e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.24e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.61e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.64e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 4.54e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.55e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.06e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.80e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 5.31e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 3.63e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.14e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 8.88e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.34e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 6.63e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.34e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 7.05e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.60e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.32e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.09e6T + 8.07e13T^{2} \) |
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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
−9.224544944444027208007223563039, −8.303054677562634656463061581979, −7.959432439839235269720378508497, −6.89130996673542315786723659986, −5.61766589043936379282993947993, −4.57206787990281507687857244626, −3.54555003240942768601610791903, −2.76626444685557140505259184262, −1.03748282858470802522320351662, 0,
1.03748282858470802522320351662, 2.76626444685557140505259184262, 3.54555003240942768601610791903, 4.57206787990281507687857244626, 5.61766589043936379282993947993, 6.89130996673542315786723659986, 7.959432439839235269720378508497, 8.303054677562634656463061581979, 9.224544944444027208007223563039
