| 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
| \(L(4)\) |
\(\approx\) |
\(3.222423852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.222423852\) |
| \(L(\frac{9}{2})\) |
|
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
−11.20309921992284709110588686200, −10.12298834799454091772777082385, −9.499407343648708737641630884452, −8.439033483227483600277800761451, −6.57794352144059805946523926504, −6.17670557980276367398790857160, −5.15184522915745120115311372330, −3.63487380941590684370454345964, −1.83638260928764503488999420973, −1.13786341934319168502901168101,
1.13786341934319168502901168101, 1.83638260928764503488999420973, 3.63487380941590684370454345964, 5.15184522915745120115311372330, 6.17670557980276367398790857160, 6.57794352144059805946523926504, 8.439033483227483600277800761451, 9.499407343648708737641630884452, 10.12298834799454091772777082385, 11.20309921992284709110588686200
