| L(s) = 1 | − 360·3-s + 2.52e4·7-s + 6.48e4·9-s − 4.55e5·13-s + 2.49e5·16-s − 9.07e6·21-s − 7.08e6·27-s − 6.91e6·31-s + 2.30e7·37-s + 1.63e8·39-s − 9.74e7·43-s − 8.98e7·48-s + 3.17e8·49-s + 6.07e7·61-s + 1.63e9·63-s − 2.03e8·67-s + 1.59e8·73-s + 5.15e8·81-s − 1.14e10·91-s + 2.49e9·93-s − 3.22e9·97-s + 5.43e7·103-s − 8.28e9·111-s + 6.29e9·112-s − 2.94e10·117-s + 9.23e9·121-s + ⋯ |
| L(s) = 1 | − 2.56·3-s + 3.96·7-s + 3.29·9-s − 4.41·13-s + 0.952·16-s − 10.1·21-s − 2.56·27-s − 1.34·31-s + 2.01·37-s + 11.3·39-s − 4.34·43-s − 2.44·48-s + 7.86·49-s + 0.561·61-s + 13.0·63-s − 1.23·67-s + 0.656·73-s + 1.33·81-s − 17.5·91-s + 3.45·93-s − 3.69·97-s + 0.0475·103-s − 5.18·111-s + 3.77·112-s − 14.5·117-s + 3.91·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.5641805798\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5641805798\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 + 40 p^{2} T + 800 p^{4} T^{2} + 40 p^{11} T^{3} + p^{18} T^{4} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - 15607 p^{4} T^{4} + p^{36} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 1800 p T + 1620000 p^{2} T^{2} - 1800 p^{10} T^{3} + p^{18} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 4615584982 T^{2} + p^{18} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 227520 T + 25882675200 T^{2} + 227520 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - \)\(15\!\cdots\!82\)\( T^{4} + p^{36} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 639840630742 T^{2} + p^{18} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - \)\(55\!\cdots\!62\)\( T^{4} + p^{36} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 2555497446662 T^{2} + p^{18} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 1729928 T + p^{9} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 11507040 T + 66205984780800 T^{2} - 11507040 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 416558667805522 T^{2} + p^{18} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 48738600 T + 1187725564980000 T^{2} + 48738600 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + \)\(18\!\cdots\!78\)\( T^{4} + p^{36} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - \)\(10\!\cdots\!22\)\( T^{4} + p^{36} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 6491019052340278 T^{2} + p^{18} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 15186242 T + p^{9} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 101529720 T + 5154142021639200 T^{2} + 101529720 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 89874898731400462 T^{2} + p^{18} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 79621920 T + 3169825072243200 T^{2} - 79621920 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 200107037864981662 T^{2} + p^{18} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + \)\(69\!\cdots\!18\)\( T^{4} + p^{36} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 127978958256992818 T^{2} + p^{18} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 1611067680 T + 1297769534770291200 T^{2} + 1611067680 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.684623119506683713137968921587, −8.368794124796850653279191210780, −7.977605827231769560078135161456, −7.926803576888979888887823498027, −7.53845928693069929073574286392, −7.33927254407786216111633480099, −6.90657326386774839877528629519, −6.84272871712439638633714577473, −6.18635062355392256392012948448, −5.64984173012284654249708046945, −5.43790114306611053328217975232, −5.25807122056850958082270608498, −4.96040054902942787911004211191, −4.82638485803227785993640959500, −4.58248016494900505826951210991, −4.38511802221405672294036168132, −3.75565583071546426657663485729, −2.85409522794290105550226289624, −2.62861816180396362912988688916, −1.87167136854507473916630458684, −1.74519845206968323086732713241, −1.66469257973600752730153532742, −0.941008055204064641353037824142, −0.56303230404389863026765954605, −0.16420147883217560514570593755,
0.16420147883217560514570593755, 0.56303230404389863026765954605, 0.941008055204064641353037824142, 1.66469257973600752730153532742, 1.74519845206968323086732713241, 1.87167136854507473916630458684, 2.62861816180396362912988688916, 2.85409522794290105550226289624, 3.75565583071546426657663485729, 4.38511802221405672294036168132, 4.58248016494900505826951210991, 4.82638485803227785993640959500, 4.96040054902942787911004211191, 5.25807122056850958082270608498, 5.43790114306611053328217975232, 5.64984173012284654249708046945, 6.18635062355392256392012948448, 6.84272871712439638633714577473, 6.90657326386774839877528629519, 7.33927254407786216111633480099, 7.53845928693069929073574286392, 7.926803576888979888887823498027, 7.977605827231769560078135161456, 8.368794124796850653279191210780, 8.684623119506683713137968921587
