| L(s) = 1 | − 300·5-s + 690·9-s + 308·13-s + 1.49e4·17-s + 3.62e4·25-s − 2.15e4·29-s − 2.27e4·37-s + 1.34e5·41-s − 2.07e5·45-s − 1.36e5·49-s + 2.19e5·53-s + 6.13e5·61-s − 9.24e4·65-s + 3.31e5·73-s − 5.53e4·81-s − 4.47e6·85-s + 9.43e5·89-s + 1.82e6·97-s − 1.91e6·101-s − 1.22e5·109-s − 5.63e6·113-s + 2.12e5·117-s + 2.70e6·121-s + 5.62e5·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 2.39·5-s + 0.946·9-s + 0.140·13-s + 3.03·17-s + 2.31·25-s − 0.882·29-s − 0.448·37-s + 1.94·41-s − 2.27·45-s − 1.15·49-s + 1.47·53-s + 2.70·61-s − 0.336·65-s + 0.851·73-s − 0.104·81-s − 7.28·85-s + 1.33·89-s + 1.99·97-s − 1.85·101-s − 0.0943·109-s − 3.90·113-s + 0.132·117-s + 1.52·121-s + 0.287·125-s + 2.11·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.186046472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.186046472\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 230 p T^{2} + p^{12} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 6 p^{2} T + p^{6} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 136414 T^{2} + p^{12} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 22370 p^{2} T^{2} + p^{12} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 154 T + p^{6} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 7458 T + p^{6} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 89538290 T^{2} + p^{12} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 224159330 T^{2} + p^{12} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 10758 T + p^{6} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1771026050 T^{2} + p^{12} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 11350 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 67122 T + p^{6} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 6312377810 T^{2} + p^{12} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 16727661506 T^{2} + p^{12} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 109962 T + p^{6} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 9193136110 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 306746 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 132365032370 T^{2} + p^{12} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 117597798434 T^{2} + p^{12} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 165682 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 94625088958 T^{2} + p^{12} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 409096407026 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 471954 T + p^{6} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 910594 T + p^{6} T^{2} )^{2} \) |
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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
−18.51424336700464162249999718126, −17.39959232044870421331741843064, −16.37880943374602554486244483832, −16.16595701881246663788934097938, −15.61908662502753862843695369547, −14.74744569566762519934132576670, −14.47282453352100109102682289760, −13.11862411773759558586211064186, −12.37193444645476470780455908480, −11.94493948360992662369585241973, −11.31227244744948381628640898034, −10.35053514129421046638436765303, −9.522930745763892907026271656484, −8.119962954048023734856443129214, −7.76478140331783440973976323845, −7.11660492002165880212406947157, −5.45358679998104018047995759692, −4.05193682631012812589353841347, −3.50829100418603698155215549268, −0.837760838626976873922181219140,
0.837760838626976873922181219140, 3.50829100418603698155215549268, 4.05193682631012812589353841347, 5.45358679998104018047995759692, 7.11660492002165880212406947157, 7.76478140331783440973976323845, 8.119962954048023734856443129214, 9.522930745763892907026271656484, 10.35053514129421046638436765303, 11.31227244744948381628640898034, 11.94493948360992662369585241973, 12.37193444645476470780455908480, 13.11862411773759558586211064186, 14.47282453352100109102682289760, 14.74744569566762519934132576670, 15.61908662502753862843695369547, 16.16595701881246663788934097938, 16.37880943374602554486244483832, 17.39959232044870421331741843064, 18.51424336700464162249999718126
