| L(s) = 1 | + 10·3-s + 18·4-s + 150·7-s + 19·9-s + 180·12-s − 110·13-s + 68·16-s − 694·19-s + 1.50e3·21-s − 620·27-s + 2.70e3·28-s − 6·31-s + 342·36-s + 4.46e3·37-s − 1.10e3·39-s + 2.95e3·43-s + 680·48-s + 1.20e4·49-s − 1.98e3·52-s − 6.94e3·57-s + 734·61-s + 2.85e3·63-s − 3.38e3·64-s + 4.47e3·67-s − 1.39e4·73-s − 1.24e4·76-s + 9.03e3·79-s + ⋯ |
| L(s) = 1 | + 10/9·3-s + 9/8·4-s + 3.06·7-s + 0.234·9-s + 5/4·12-s − 0.650·13-s + 0.265·16-s − 1.92·19-s + 3.40·21-s − 0.850·27-s + 3.44·28-s − 0.00624·31-s + 0.263·36-s + 3.25·37-s − 0.723·39-s + 1.59·43-s + 0.295·48-s + 5.02·49-s − 0.732·52-s − 2.13·57-s + 0.197·61-s + 0.718·63-s − 0.826·64-s + 0.995·67-s − 2.61·73-s − 2.16·76-s + 1.44·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(5.651968813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.651968813\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - 10 T + p^{4} T^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 9 p T^{2} + p^{8} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 75 T + p^{4} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 27882 T^{2} + p^{8} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 55 T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 84342 T^{2} + p^{8} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 347 T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 135818 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 673962 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2230 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 778122 T^{2} + p^{8} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 1475 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 6315138 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 15482538 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 16148322 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 367 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2235 T + p^{4} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 50586762 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6970 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4518 T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 94817858 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 60166082 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 4535 T + p^{4} T^{2} )^{2} \) |
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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
−14.42880650280369702937370319888, −13.74822153413072631232695180821, −13.03449903429706853470273871136, −12.36883495789767616604530773653, −11.60050398920785896442234654403, −11.29939266043791078290732660389, −10.97592737591443321249001447853, −10.38293414316757110702257325765, −9.381859529590001715471576252526, −8.718534992948701383428460289898, −8.193452280767307438520070313760, −7.74258748084810209646350084946, −7.44368427291958506464570618509, −6.37712175886160370647477953374, −5.56594500774308635946513231440, −4.54526885367438370799698105702, −4.23028773535159071515574944183, −2.39003161505661136108471691398, −2.37956365583071556885525405986, −1.36856840038270491048893858194,
1.36856840038270491048893858194, 2.37956365583071556885525405986, 2.39003161505661136108471691398, 4.23028773535159071515574944183, 4.54526885367438370799698105702, 5.56594500774308635946513231440, 6.37712175886160370647477953374, 7.44368427291958506464570618509, 7.74258748084810209646350084946, 8.193452280767307438520070313760, 8.718534992948701383428460289898, 9.381859529590001715471576252526, 10.38293414316757110702257325765, 10.97592737591443321249001447853, 11.29939266043791078290732660389, 11.60050398920785896442234654403, 12.36883495789767616604530773653, 13.03449903429706853470273871136, 13.74822153413072631232695180821, 14.42880650280369702937370319888
