| L(s) = 1 | + 42.5·3-s + 314.·5-s − 84.3·7-s − 372.·9-s + 8.31e3·11-s − 7.32e3·13-s + 1.34e4·15-s + 2.07e4·17-s + 2.92e4·19-s − 3.59e3·21-s + 7.54e4·23-s + 2.09e4·25-s − 1.09e5·27-s − 2.33e5·29-s − 2.37e5·31-s + 3.54e5·33-s − 2.65e4·35-s − 1.50e5·37-s − 3.11e5·39-s + 2.77e5·41-s − 2.86e5·43-s − 1.17e5·45-s − 6.76e4·47-s − 8.16e5·49-s + 8.84e5·51-s − 2.61e5·53-s + 2.61e6·55-s + ⋯ |
| L(s) = 1 | + 0.910·3-s + 1.12·5-s − 0.0929·7-s − 0.170·9-s + 1.88·11-s − 0.924·13-s + 1.02·15-s + 1.02·17-s + 0.977·19-s − 0.0846·21-s + 1.29·23-s + 0.268·25-s − 1.06·27-s − 1.78·29-s − 1.42·31-s + 1.71·33-s − 0.104·35-s − 0.486·37-s − 0.841·39-s + 0.629·41-s − 0.550·43-s − 0.191·45-s − 0.0950·47-s − 0.991·49-s + 0.933·51-s − 0.240·53-s + 2.12·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.744377984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.744377984\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 42.5T + 2.18e3T^{2} \) |
| 5 | \( 1 - 314.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 84.3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 8.31e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 7.32e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.07e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.92e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 7.54e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.33e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.37e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.50e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.77e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.86e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 6.76e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.61e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.04e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.39e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.26e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.53e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.10e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.34e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.19e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 8.90e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.81e6T + 8.07e13T^{2} \) |
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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
−14.61497654010218883004986059732, −14.38631876057852652802652105269, −13.06905078700878385277560044015, −11.57723033509543454458477880890, −9.636384567023257739496827117613, −9.120037133683547557325872742938, −7.26625891594721368659998847687, −5.60605536311888276927223367927, −3.36943075279122854187329717700, −1.67185844863357814779413927793,
1.67185844863357814779413927793, 3.36943075279122854187329717700, 5.60605536311888276927223367927, 7.26625891594721368659998847687, 9.120037133683547557325872742938, 9.636384567023257739496827117613, 11.57723033509543454458477880890, 13.06905078700878385277560044015, 14.38631876057852652802652105269, 14.61497654010218883004986059732
