| L(s) = 1 | − 16·5-s − 12·7-s + 448·11-s + 206·13-s + 1.95e3·17-s + 1.06e3·19-s − 3.71e3·23-s − 2.86e3·25-s − 4.08e3·29-s − 5.32e3·31-s + 192·35-s + 9.69e3·37-s − 9.12e3·41-s + 1.65e4·43-s + 1.42e4·47-s − 1.66e4·49-s + 2.17e4·53-s − 7.16e3·55-s − 3.16e4·59-s + 1.31e4·61-s − 3.29e3·65-s + 2.70e4·67-s − 9.72e3·71-s + 9.04e3·73-s − 5.37e3·77-s + 5.82e4·79-s + 8.63e4·83-s + ⋯ |
| L(s) = 1 | − 0.286·5-s − 0.0925·7-s + 1.11·11-s + 0.338·13-s + 1.63·17-s + 0.676·19-s − 1.46·23-s − 0.918·25-s − 0.900·29-s − 0.995·31-s + 0.0264·35-s + 1.16·37-s − 0.847·41-s + 1.36·43-s + 0.938·47-s − 0.991·49-s + 1.06·53-s − 0.319·55-s − 1.18·59-s + 0.452·61-s − 0.0967·65-s + 0.736·67-s − 0.229·71-s + 0.198·73-s − 0.103·77-s + 1.05·79-s + 1.37·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.212552712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.212552712\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 16 T + p^{5} T^{2} \) |
| 7 | \( 1 + 12 T + p^{5} T^{2} \) |
| 11 | \( 1 - 448 T + p^{5} T^{2} \) |
| 13 | \( 1 - 206 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1952 T + p^{5} T^{2} \) |
| 19 | \( 1 - 56 p T + p^{5} T^{2} \) |
| 23 | \( 1 + 3712 T + p^{5} T^{2} \) |
| 29 | \( 1 + 4080 T + p^{5} T^{2} \) |
| 31 | \( 1 + 5324 T + p^{5} T^{2} \) |
| 37 | \( 1 - 9690 T + p^{5} T^{2} \) |
| 41 | \( 1 + 9120 T + p^{5} T^{2} \) |
| 43 | \( 1 - 16552 T + p^{5} T^{2} \) |
| 47 | \( 1 - 14208 T + p^{5} T^{2} \) |
| 53 | \( 1 - 21776 T + p^{5} T^{2} \) |
| 59 | \( 1 + 31616 T + p^{5} T^{2} \) |
| 61 | \( 1 - 13154 T + p^{5} T^{2} \) |
| 67 | \( 1 - 27056 T + p^{5} T^{2} \) |
| 71 | \( 1 + 9728 T + p^{5} T^{2} \) |
| 73 | \( 1 - 9046 T + p^{5} T^{2} \) |
| 79 | \( 1 - 58292 T + p^{5} T^{2} \) |
| 83 | \( 1 - 86336 T + p^{5} T^{2} \) |
| 89 | \( 1 - 75072 T + p^{5} T^{2} \) |
| 97 | \( 1 - 76046 T + p^{5} T^{2} \) |
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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
−9.774940737038852627290506065465, −9.249344378451523081197662654265, −8.009206451989903635376748179052, −7.44138371855339463870619853684, −6.19885751684129398098845515734, −5.50393652793077513574683061491, −4.04441243836929367360463490271, −3.45433408276138770835333790800, −1.88153024640212573376818252450, −0.74984680301218209574803978867,
0.74984680301218209574803978867, 1.88153024640212573376818252450, 3.45433408276138770835333790800, 4.04441243836929367360463490271, 5.50393652793077513574683061491, 6.19885751684129398098845515734, 7.44138371855339463870619853684, 8.009206451989903635376748179052, 9.249344378451523081197662654265, 9.774940737038852627290506065465
