| L(s) = 1 | + 27·3-s + 223.·5-s − 1.52e3·7-s + 729·9-s − 1.60e3·11-s − 2.95e3·13-s + 6.02e3·15-s − 2.01e4·17-s − 7.74e3·19-s − 4.12e4·21-s + 7.14e4·23-s − 2.83e4·25-s + 1.96e4·27-s − 2.51e4·29-s + 2.70e5·31-s − 4.33e4·33-s − 3.40e5·35-s − 3.94e5·37-s − 7.99e4·39-s + 1.81e5·41-s + 5.55e5·43-s + 1.62e5·45-s + 6.69e5·47-s + 1.50e6·49-s − 5.43e5·51-s + 7.72e5·53-s − 3.57e5·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.798·5-s − 1.68·7-s + 0.333·9-s − 0.363·11-s − 0.373·13-s + 0.460·15-s − 0.994·17-s − 0.259·19-s − 0.971·21-s + 1.22·23-s − 0.362·25-s + 0.192·27-s − 0.191·29-s + 1.63·31-s − 0.209·33-s − 1.34·35-s − 1.27·37-s − 0.215·39-s + 0.411·41-s + 1.06·43-s + 0.266·45-s + 0.941·47-s + 1.82·49-s − 0.574·51-s + 0.713·53-s − 0.290·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.173863572\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.173863572\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 27T \) |
| good | 5 | \( 1 - 223.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.52e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.60e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 2.95e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.01e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 7.74e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 7.14e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.51e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.70e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.94e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.81e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.55e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 6.69e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 7.72e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.30e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.68e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.08e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 8.59e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.61e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.33e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.43e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.18e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 4.05e6T + 8.07e13T^{2} \) |
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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
−9.952829354571577916300459798541, −9.346696195874927909521855541252, −8.552395149026283792557522590738, −7.14227645127912153903585133569, −6.50122831582392207659706877143, −5.46705011172647851440252742724, −4.11088868249971280108505285318, −2.92571462874581579605513022456, −2.25089198770484563996430673505, −0.63709261949317737701380822050,
0.63709261949317737701380822050, 2.25089198770484563996430673505, 2.92571462874581579605513022456, 4.11088868249971280108505285318, 5.46705011172647851440252742724, 6.50122831582392207659706877143, 7.14227645127912153903585133569, 8.552395149026283792557522590738, 9.346696195874927909521855541252, 9.952829354571577916300459798541
