| L(s) = 1 | + 530·5-s + 120·7-s + 7.19e3·11-s − 9.62e3·13-s − 1.86e4·17-s + 7.00e3·19-s + 6.37e4·23-s + 2.02e5·25-s − 2.93e4·29-s + 8.79e4·31-s + 6.36e4·35-s + 2.27e5·37-s + 1.60e5·41-s + 1.36e5·43-s + 1.20e6·47-s − 8.09e5·49-s + 3.98e5·53-s + 3.81e6·55-s − 1.15e6·59-s − 2.07e6·61-s − 5.10e6·65-s − 4.07e6·67-s + 3.83e5·71-s + 3.00e6·73-s + 8.63e5·77-s − 4.94e6·79-s + 9.16e6·83-s + ⋯ |
| L(s) = 1 | + 1.89·5-s + 0.132·7-s + 1.63·11-s − 1.21·13-s − 0.921·17-s + 0.234·19-s + 1.09·23-s + 2.59·25-s − 0.223·29-s + 0.530·31-s + 0.250·35-s + 0.739·37-s + 0.364·41-s + 0.261·43-s + 1.69·47-s − 0.982·49-s + 0.367·53-s + 3.09·55-s − 0.730·59-s − 1.16·61-s − 2.30·65-s − 1.65·67-s + 0.127·71-s + 0.904·73-s + 0.215·77-s − 1.12·79-s + 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.875321633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.875321633\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 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
−13.35352748928501130390257633550, −12.20367613295835049485277343534, −10.82077528728141577715674614034, −9.557439658168306676895304459846, −9.070241704624324375078782566034, −6.99253387010340039610348370574, −6.02013762868646847984162774789, −4.68509182158740914822515015732, −2.54792238525638176275401508653, −1.30318717545784188015232809520,
1.30318717545784188015232809520, 2.54792238525638176275401508653, 4.68509182158740914822515015732, 6.02013762868646847984162774789, 6.99253387010340039610348370574, 9.070241704624324375078782566034, 9.557439658168306676895304459846, 10.82077528728141577715674614034, 12.20367613295835049485277343534, 13.35352748928501130390257633550
