| L(s) = 1 | + 27·3-s + 127.·5-s − 547.·7-s + 729·9-s + 5.22e3·11-s + 1.10e4·13-s + 3.43e3·15-s + 3.56e3·17-s + 3.78e4·19-s − 1.47e4·21-s + 3.18e3·23-s − 6.19e4·25-s + 1.96e4·27-s − 3.65e3·29-s − 1.81e5·31-s + 1.41e5·33-s − 6.96e4·35-s − 1.33e5·37-s + 2.97e5·39-s − 1.89e5·41-s + 4.33e5·43-s + 9.28e4·45-s − 1.27e5·47-s − 5.24e5·49-s + 9.61e4·51-s + 1.84e6·53-s + 6.65e5·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.455·5-s − 0.603·7-s + 0.333·9-s + 1.18·11-s + 1.39·13-s + 0.262·15-s + 0.175·17-s + 1.26·19-s − 0.348·21-s + 0.0546·23-s − 0.792·25-s + 0.192·27-s − 0.0278·29-s − 1.09·31-s + 0.683·33-s − 0.274·35-s − 0.434·37-s + 0.802·39-s − 0.430·41-s + 0.831·43-s + 0.151·45-s − 0.179·47-s − 0.636·49-s + 0.101·51-s + 1.70·53-s + 0.539·55-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
| \(L(4)\) |
\(\approx\) |
\(3.611514634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.611514634\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 127.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 547.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.22e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.10e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.56e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.78e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.18e3T + 3.40e9T^{2} \) |
| 29 | \( 1 + 3.65e3T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.81e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.33e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.89e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.33e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.27e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.84e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 4.49e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 3.57e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.13e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.53e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.45e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.20e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.08e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 9.79e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + 4.99e6T + 8.07e13T^{2} \) |
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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
−9.880295071976344859725421323037, −9.286930035921115169567144135322, −8.497442047298057970554776763462, −7.31020520043315155868491829700, −6.38482143244802393387451862830, −5.50809039923203088895995085565, −3.91753541222357056912590488251, −3.31463360952451586270682800170, −1.87266338293512796392613730885, −0.910401733167495429244790117976,
0.910401733167495429244790117976, 1.87266338293512796392613730885, 3.31463360952451586270682800170, 3.91753541222357056912590488251, 5.50809039923203088895995085565, 6.38482143244802393387451862830, 7.31020520043315155868491829700, 8.497442047298057970554776763462, 9.286930035921115169567144135322, 9.880295071976344859725421323037
