| L(s) = 1 | − 90·3-s + 3.50e3·7-s + 1.53e3·9-s + 5.14e4·13-s − 3.78e4·19-s − 3.15e5·21-s + 7.30e5·25-s + 4.51e5·27-s + 7.02e5·31-s + 2.67e6·37-s − 4.63e6·39-s + 7.05e6·43-s − 2.34e6·49-s + 3.40e6·57-s + 1.50e6·61-s + 5.38e6·63-s − 4.53e6·67-s + 5.53e7·73-s − 6.57e7·75-s + 4.59e7·79-s − 5.07e7·81-s + 1.80e8·91-s − 6.32e7·93-s + 2.94e8·97-s + 3.32e8·103-s − 2.19e8·109-s − 2.40e8·111-s + ⋯ |
| L(s) = 1 | − 1.11·3-s + 1.45·7-s + 0.234·9-s + 1.80·13-s − 0.290·19-s − 1.61·21-s + 1.87·25-s + 0.850·27-s + 0.761·31-s + 1.42·37-s − 2.00·39-s + 2.06·43-s − 0.406·49-s + 0.322·57-s + 0.108·61-s + 0.341·63-s − 0.225·67-s + 1.94·73-s − 2.07·75-s + 1.18·79-s − 1.17·81-s + 2.62·91-s − 0.845·93-s + 3.32·97-s + 2.95·103-s − 1.55·109-s − 1.58·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.732724957\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.732724957\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + 10 p^{2} T + p^{8} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 29234 p^{2} T^{2} + p^{16} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 250 p T + p^{8} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 380283362 T^{2} + p^{16} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 25730 T + p^{8} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 8342551298 T^{2} + p^{16} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 18938 T + p^{8} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 64711613182 T^{2} + p^{16} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 788066452322 T^{2} + p^{16} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11338 p T + p^{8} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 1335170 T + p^{8} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 12452468931842 T^{2} + p^{16} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 3526150 T + p^{8} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30967680304898 T^{2} + p^{16} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 80936075395298 T^{2} + p^{16} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 105562517046242 T^{2} + p^{16} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 753602 T + p^{8} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 2268890 T + p^{8} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 1001758688017922 T^{2} + p^{16} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 27672770 T + p^{8} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 22980982 T + p^{8} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 2352070843223138 T^{2} + p^{16} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2600204109557762 T^{2} + p^{16} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 147271010 T + p^{8} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.27448520457435243076198718689, −13.70260107437466586894635630160, −12.86274930643841808110116366317, −12.51277768015357918966946082947, −11.46640651235785166298089456403, −11.43264430915779336999848573295, −10.76121746546170846429735256890, −10.51076159371345303411999712944, −9.310256699504618723072463268847, −8.612045762433163807982767093047, −8.172804267250429017577728294865, −7.36919042493287258435336054758, −6.20578692927325601483921042856, −6.17716237888234916682840836213, −4.98065529848868112208831702534, −4.68812500496455518655213585337, −3.61288252650347321324123224000, −2.40001692032778338434242710382, −1.15754116742170621321686701466, −0.817325643783789336294085270175,
0.817325643783789336294085270175, 1.15754116742170621321686701466, 2.40001692032778338434242710382, 3.61288252650347321324123224000, 4.68812500496455518655213585337, 4.98065529848868112208831702534, 6.17716237888234916682840836213, 6.20578692927325601483921042856, 7.36919042493287258435336054758, 8.172804267250429017577728294865, 8.612045762433163807982767093047, 9.310256699504618723072463268847, 10.51076159371345303411999712944, 10.76121746546170846429735256890, 11.43264430915779336999848573295, 11.46640651235785166298089456403, 12.51277768015357918966946082947, 12.86274930643841808110116366317, 13.70260107437466586894635630160, 14.27448520457435243076198718689
