| L(s) = 1 | + 3.52e3·4-s + 2.90e5·9-s + 1.06e6·11-s + 8.19e6·16-s − 2.13e7·19-s − 2.56e8·29-s − 1.05e8·31-s + 1.02e9·36-s + 6.16e8·41-s + 3.76e9·44-s + 3.67e9·49-s + 1.03e10·59-s + 1.39e10·61-s + 1.40e10·64-s + 1.95e10·71-s − 7.50e10·76-s − 7.62e10·79-s + 5.31e10·81-s + 4.99e10·89-s + 3.10e11·99-s + 1.63e11·101-s − 1.46e11·109-s − 9.03e11·116-s + 2.86e11·121-s − 3.72e11·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1.71·4-s + 1.64·9-s + 2.00·11-s + 1.95·16-s − 1.97·19-s − 2.32·29-s − 0.663·31-s + 2.82·36-s + 0.830·41-s + 3.44·44-s + 1.85·49-s + 1.88·59-s + 2.10·61-s + 1.63·64-s + 1.28·71-s − 3.39·76-s − 2.78·79-s + 1.69·81-s + 0.948·89-s + 3.28·99-s + 1.54·101-s − 0.914·109-s − 3.99·116-s + 1.00·121-s − 1.13·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(5.842565863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.842565863\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 55 p^{6} T^{2} + p^{22} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 3590 p^{4} T^{2} + p^{22} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 74985550 p^{2} T^{2} + p^{22} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 534612 T + p^{11} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3250539591430 T^{2} + p^{22} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 20851868202910 T^{2} + p^{22} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 10661420 T + p^{11} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1558047924961870 T^{2} + p^{22} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 128406630 T + p^{11} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 52843168 T + p^{11} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 322633551760058230 T^{2} + p^{22} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 308120442 T + p^{11} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 1858294189067944150 T^{2} + p^{22} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 2277523508785437410 T^{2} + p^{22} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 15990678067626115990 T^{2} + p^{22} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 5189203740 T + p^{11} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6956478662 T + p^{11} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 4573302143790884710 T^{2} + p^{22} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 9791485272 T + p^{11} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - \)\(62\!\cdots\!70\)\( T^{2} + p^{22} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 38116845680 T + p^{11} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - \)\(17\!\cdots\!10\)\( T^{2} + p^{22} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 24992917110 T + p^{11} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - \)\(86\!\cdots\!90\)\( T^{2} + p^{22} T^{4} \) |
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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
−15.29855715295858882746184393866, −14.75608454664552040295093498462, −14.46103779284686886134617321896, −13.02665133579539324773727066782, −12.82177103312229604885240164752, −12.00134958167696348120128666180, −11.41590413706798709834114318567, −10.87452484069808015158852025020, −10.17759706355169179010704376730, −9.431828185380058711508647246190, −8.605203624331552523342271162593, −7.39440585809827607424925747760, −7.02200768285304446621713590224, −6.48042515491736724143788439794, −5.68996044964596848995831986223, −4.04256954016969285500285515474, −3.84792282123345933351020564158, −2.18443434866471447501051418763, −1.80473776267650033488740826688, −0.926385349266899633125817506920,
0.926385349266899633125817506920, 1.80473776267650033488740826688, 2.18443434866471447501051418763, 3.84792282123345933351020564158, 4.04256954016969285500285515474, 5.68996044964596848995831986223, 6.48042515491736724143788439794, 7.02200768285304446621713590224, 7.39440585809827607424925747760, 8.605203624331552523342271162593, 9.431828185380058711508647246190, 10.17759706355169179010704376730, 10.87452484069808015158852025020, 11.41590413706798709834114318567, 12.00134958167696348120128666180, 12.82177103312229604885240164752, 13.02665133579539324773727066782, 14.46103779284686886134617321896, 14.75608454664552040295093498462, 15.29855715295858882746184393866
