| L(s) = 1 | + 18·3-s + 25·5-s + 242·7-s + 81·9-s − 656·11-s + 206·13-s + 450·15-s + 1.69e3·17-s + 1.36e3·19-s + 4.35e3·21-s + 2.19e3·23-s + 625·25-s − 2.91e3·27-s + 2.21e3·29-s − 1.70e3·31-s − 1.18e4·33-s + 6.05e3·35-s + 846·37-s + 3.70e3·39-s − 1.81e3·41-s − 1.05e4·43-s + 2.02e3·45-s + 1.20e4·47-s + 4.17e4·49-s + 3.04e4·51-s − 3.25e4·53-s − 1.64e4·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 1.86·7-s + 1/3·9-s − 1.63·11-s + 0.338·13-s + 0.516·15-s + 1.41·17-s + 0.866·19-s + 2.15·21-s + 0.866·23-s + 1/5·25-s − 0.769·27-s + 0.489·29-s − 0.317·31-s − 1.88·33-s + 0.834·35-s + 0.101·37-s + 0.390·39-s − 0.168·41-s − 0.868·43-s + 0.149·45-s + 0.797·47-s + 2.48·49-s + 1.63·51-s − 1.59·53-s − 0.731·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.254417050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.254417050\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 - 2 p^{2} T + p^{5} T^{2} \) |
| 7 | \( 1 - 242 T + p^{5} T^{2} \) |
| 11 | \( 1 + 656 T + p^{5} T^{2} \) |
| 13 | \( 1 - 206 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1690 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1364 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2198 T + p^{5} T^{2} \) |
| 29 | \( 1 - 2218 T + p^{5} T^{2} \) |
| 31 | \( 1 + 1700 T + p^{5} T^{2} \) |
| 37 | \( 1 - 846 T + p^{5} T^{2} \) |
| 41 | \( 1 + 1818 T + p^{5} T^{2} \) |
| 43 | \( 1 + 10534 T + p^{5} T^{2} \) |
| 47 | \( 1 - 12074 T + p^{5} T^{2} \) |
| 53 | \( 1 + 32586 T + p^{5} T^{2} \) |
| 59 | \( 1 + 8668 T + p^{5} T^{2} \) |
| 61 | \( 1 - 34670 T + p^{5} T^{2} \) |
| 67 | \( 1 - 47566 T + p^{5} T^{2} \) |
| 71 | \( 1 - 948 T + p^{5} T^{2} \) |
| 73 | \( 1 + 63102 T + p^{5} T^{2} \) |
| 79 | \( 1 - 46536 T + p^{5} T^{2} \) |
| 83 | \( 1 - 88778 T + p^{5} T^{2} \) |
| 89 | \( 1 + 104934 T + p^{5} T^{2} \) |
| 97 | \( 1 + 36254 T + p^{5} T^{2} \) |
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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
−10.73961229098923815250942942827, −9.817860805182915459001517915963, −8.671870254695737220960948895428, −7.994226328434341816491560854666, −7.47753649433391359699133553056, −5.53183030072664567540848601680, −4.92689238640077636321922521207, −3.30561729592740628816169489817, −2.31379691433122749455578954394, −1.20918872506854538701467737797,
1.20918872506854538701467737797, 2.31379691433122749455578954394, 3.30561729592740628816169489817, 4.92689238640077636321922521207, 5.53183030072664567540848601680, 7.47753649433391359699133553056, 7.994226328434341816491560854666, 8.671870254695737220960948895428, 9.817860805182915459001517915963, 10.73961229098923815250942942827
