| L(s) = 1 | + 6.13e5·3-s − 8.38e6·4-s + 2.82e11·9-s − 5.14e12·12-s + 7.03e13·16-s + 1.66e15·19-s − 2.38e16·25-s + 1.15e17·27-s − 2.36e18·36-s + 2.16e19·43-s + 4.31e19·48-s + 5.47e19·49-s + 1.02e21·57-s − 5.90e20·64-s + 3.99e21·67-s − 4.60e21·73-s − 1.46e22·75-s − 1.39e22·76-s + 4.42e22·81-s − 8.84e22·97-s + 1.99e23·100-s − 9.68e23·108-s + 2.79e23·121-s + 127-s + 1.32e25·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.99·3-s − 4-s + 2.99·9-s − 1.99·12-s + 16-s + 3.27·19-s − 2·25-s + 3.99·27-s − 2.99·36-s + 3.55·43-s + 1.99·48-s + 2·49-s + 6.54·57-s − 64-s + 3.99·67-s − 1.71·73-s − 3.99·75-s − 3.27·76-s + 4.99·81-s − 1.25·97-s + 2·100-s − 3.99·108-s + 0.312·121-s + 7.11·129-s + 2.99·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+23/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(\approx\) |
\(9.326430022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.326430022\) |
| \(L(\frac{25}{2})\) |
|
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 | $C_2$ | \( 1 + p^{23} T^{2} \) |
| 3 | $C_2$ | \( 1 - 613546 T + p^{23} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p^{23} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p^{23} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 1438967473134 T + p^{23} T^{2} )( 1 + 1438967473134 T + p^{23} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - p^{23} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 30613766608038 T + p^{23} T^{2} )( 1 + 30613766608038 T + p^{23} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 831457853116598 T + p^{23} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{23} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{23} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p^{23} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p^{23} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6878399708800712406 T + p^{23} T^{2} )( 1 + 6878399708800712406 T + p^{23} T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10834233708543928610 T + p^{23} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{23} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{23} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 90261986017063503246 T + p^{23} T^{2} )( 1 + 90261986017063503246 T + p^{23} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p^{23} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - \)\(19\!\cdots\!82\)\( T + p^{23} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{23} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + \)\(23\!\cdots\!66\)\( T + p^{23} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p^{23} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - \)\(21\!\cdots\!94\)\( T + p^{23} T^{2} )( 1 + \)\(21\!\cdots\!94\)\( T + p^{23} T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - \)\(39\!\cdots\!42\)\( T + p^{23} T^{2} )( 1 + \)\(39\!\cdots\!42\)\( T + p^{23} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + \)\(44\!\cdots\!10\)\( T + p^{23} T^{2} )^{2} \) |
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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
−13.51848047559847803128529432261, −12.50181244255342217126381821483, −12.13271802985805164721670640912, −11.09115401061510685472876616466, −9.914268231008644690934473549589, −9.812659530471824993039386864530, −9.243264393763290056997051076354, −8.771191519943091871688225173568, −7.85191140165295961985812689586, −7.65837023315489244447376456220, −7.11725437907294010657781198329, −5.73497292810398597203487789467, −5.23976049856251004998497886196, −4.10416392794852022036294104365, −3.97884464298212653038880496035, −3.21380685645898761436805240646, −2.65112835427056016105394334937, −1.91562943681620696346464453242, −0.969084180387621579052594829267, −0.807204875254373402346595818980,
0.807204875254373402346595818980, 0.969084180387621579052594829267, 1.91562943681620696346464453242, 2.65112835427056016105394334937, 3.21380685645898761436805240646, 3.97884464298212653038880496035, 4.10416392794852022036294104365, 5.23976049856251004998497886196, 5.73497292810398597203487789467, 7.11725437907294010657781198329, 7.65837023315489244447376456220, 7.85191140165295961985812689586, 8.771191519943091871688225173568, 9.243264393763290056997051076354, 9.812659530471824993039386864530, 9.914268231008644690934473549589, 11.09115401061510685472876616466, 12.13271802985805164721670640912, 12.50181244255342217126381821483, 13.51848047559847803128529432261
