| L(s) = 1 | − 1.45e3·3-s − 5.03e3·4-s + 1.59e6·9-s + 7.33e6·12-s − 9.65e6·13-s + 8.55e6·16-s + 4.74e8·25-s − 1.54e9·27-s − 8.02e9·36-s + 1.40e10·39-s − 2.11e10·43-s − 1.24e10·48-s + 2.76e10·49-s + 4.85e10·52-s − 1.78e11·61-s + 4.13e10·64-s − 6.91e11·75-s − 3.43e11·79-s + 1.41e12·81-s − 2.38e12·100-s − 4.75e11·103-s + 7.79e12·108-s − 1.53e13·117-s + 4.06e12·121-s + 127-s + 3.08e13·129-s + 131-s + ⋯ |
| L(s) = 1 | − 2·3-s − 1.22·4-s + 3·9-s + 2.45·12-s − 2·13-s + 0.509·16-s + 1.94·25-s − 4·27-s − 3.68·36-s + 4·39-s − 3.35·43-s − 1.01·48-s + 2·49-s + 2.45·52-s − 3.46·61-s + 0.602·64-s − 3.88·75-s − 1.41·79-s + 5·81-s − 2.38·100-s − 0.398·103-s + 4.91·108-s − 6·117-s + 1.29·121-s + 6.70·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+6)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.1376768218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1376768218\) |
| \(L(7)\) |
|
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 | $C_1$ | \( ( 1 + p^{6} T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + p^{6} T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 5033 T^{2} + p^{24} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 474278686 T^{2} + p^{24} T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 4067560976542 T^{2} + p^{24} T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 45027555279429176062 T^{2} + p^{24} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 10591541998 T + p^{12} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + \)\(14\!\cdots\!18\)\( T^{2} + p^{24} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 59 | $C_2^2$ | \( 1 + \)\(26\!\cdots\!38\)\( T^{2} + p^{24} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 89162579378 T + p^{12} T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 71 | $C_2^2$ | \( 1 + \)\(22\!\cdots\!18\)\( T^{2} + p^{24} T^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 171789411458 T + p^{12} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + \)\(18\!\cdots\!78\)\( T^{2} + p^{24} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + \)\(36\!\cdots\!58\)\( T^{2} + p^{24} T^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
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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
−13.55165250464170730532729874081, −13.19260843090129948748249773381, −12.30731221141754498718317247144, −12.30557312710696607150863961716, −11.58787572983079751219724105560, −10.71857870467492197434295863191, −10.30254682976884212002871742971, −9.703673497826344424594144012646, −9.173218576079113786700177962736, −8.214998648690614704227033067217, −7.09233273161677456149724519199, −6.99562119105029505232655167511, −5.90448407559121244661533446905, −5.21377518113540410575297574537, −4.61546186157517552424234062174, −4.52928592003884603046815540479, −3.17786752919831671496443168804, −1.84050388713390287415492802443, −0.908149463970558112039143827185, −0.17031175159339689931415991313,
0.17031175159339689931415991313, 0.908149463970558112039143827185, 1.84050388713390287415492802443, 3.17786752919831671496443168804, 4.52928592003884603046815540479, 4.61546186157517552424234062174, 5.21377518113540410575297574537, 5.90448407559121244661533446905, 6.99562119105029505232655167511, 7.09233273161677456149724519199, 8.214998648690614704227033067217, 9.173218576079113786700177962736, 9.703673497826344424594144012646, 10.30254682976884212002871742971, 10.71857870467492197434295863191, 11.58787572983079751219724105560, 12.30557312710696607150863961716, 12.30731221141754498718317247144, 13.19260843090129948748249773381, 13.55165250464170730532729874081
