| L(s) = 1 | + 27·3-s + 530·5-s − 120·7-s + 729·9-s − 7.19e3·11-s + 9.62e3·13-s + 1.43e4·15-s + 1.86e4·17-s + 7.00e3·19-s − 3.24e3·21-s + 6.37e4·23-s + 2.02e5·25-s + 1.96e4·27-s − 2.93e4·29-s − 8.79e4·31-s − 1.94e5·33-s − 6.36e4·35-s − 2.27e5·37-s + 2.59e5·39-s − 1.60e5·41-s + 1.36e5·43-s + 3.86e5·45-s + 1.20e6·47-s − 8.09e5·49-s + 5.04e5·51-s + 3.98e5·53-s − 3.81e6·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.89·5-s − 0.132·7-s + 1/3·9-s − 1.63·11-s + 1.21·13-s + 1.09·15-s + 0.921·17-s + 0.234·19-s − 0.0763·21-s + 1.09·23-s + 2.59·25-s + 0.192·27-s − 0.223·29-s − 0.530·31-s − 0.941·33-s − 0.250·35-s − 0.739·37-s + 0.701·39-s − 0.364·41-s + 0.261·43-s + 0.632·45-s + 1.69·47-s − 0.982·49-s + 0.532·51-s + 0.367·53-s − 3.09·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.961727352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.961727352\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{3} T \) |
| good | 5 | \( 1 - 106 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 120 T + p^{7} T^{2} \) |
| 11 | \( 1 + 7196 T + p^{7} T^{2} \) |
| 13 | \( 1 - 9626 T + p^{7} T^{2} \) |
| 17 | \( 1 - 18674 T + p^{7} T^{2} \) |
| 19 | \( 1 - 7004 T + p^{7} T^{2} \) |
| 23 | \( 1 - 63704 T + p^{7} T^{2} \) |
| 29 | \( 1 + 29334 T + p^{7} T^{2} \) |
| 31 | \( 1 + 87968 T + p^{7} T^{2} \) |
| 37 | \( 1 + 227982 T + p^{7} T^{2} \) |
| 41 | \( 1 + 160806 T + p^{7} T^{2} \) |
| 43 | \( 1 - 136132 T + p^{7} T^{2} \) |
| 47 | \( 1 - 25680 p T + p^{7} T^{2} \) |
| 53 | \( 1 - 398786 T + p^{7} T^{2} \) |
| 59 | \( 1 - 1152436 T + p^{7} T^{2} \) |
| 61 | \( 1 - 2070602 T + p^{7} T^{2} \) |
| 67 | \( 1 + 4073428 T + p^{7} T^{2} \) |
| 71 | \( 1 - 383752 T + p^{7} T^{2} \) |
| 73 | \( 1 - 3006010 T + p^{7} T^{2} \) |
| 79 | \( 1 - 4948112 T + p^{7} T^{2} \) |
| 83 | \( 1 + 9163492 T + p^{7} T^{2} \) |
| 89 | \( 1 - 7304106 T + p^{7} T^{2} \) |
| 97 | \( 1 + 690526 T + p^{7} T^{2} \) |
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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
−10.83724578266208217590090512265, −10.21427492023559114845001028060, −9.315140378578810584145143210961, −8.419109278781176647790203752538, −7.12356873224319368343957411400, −5.84378715039191317829639058581, −5.18345276052336464402828707986, −3.24696170359840187547978972700, −2.27186690802170275350373087132, −1.12969626721078847454744870579,
1.12969626721078847454744870579, 2.27186690802170275350373087132, 3.24696170359840187547978972700, 5.18345276052336464402828707986, 5.84378715039191317829639058581, 7.12356873224319368343957411400, 8.419109278781176647790203752538, 9.315140378578810584145143210961, 10.21427492023559114845001028060, 10.83724578266208217590090512265
