| L(s) = 1 | + 3-s − 4-s − 2·7-s − 12-s + 2·13-s − 2·19-s − 2·21-s + 4·25-s + 2·28-s + 2·37-s + 2·39-s − 3·43-s + 49-s − 2·52-s − 2·57-s + 2·61-s − 2·67-s + 2·73-s + 4·75-s + 2·76-s − 3·79-s + 2·84-s − 4·91-s + 3·97-s − 4·100-s + 3·103-s + 3·109-s + ⋯ |
| L(s) = 1 | + 3-s − 4-s − 2·7-s − 12-s + 2·13-s − 2·19-s − 2·21-s + 4·25-s + 2·28-s + 2·37-s + 2·39-s − 3·43-s + 49-s − 2·52-s − 2·57-s + 2·61-s − 2·67-s + 2·73-s + 4·75-s + 2·76-s − 3·79-s + 2·84-s − 4·91-s + 3·97-s − 4·100-s + 3·103-s + 3·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.123191507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.123191507\) |
| \(L(1)\) |
|
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 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 31 | | \( 1 \) |
| good | 2 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 7 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 11 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 13 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 17 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 19 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 23 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 29 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 37 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 41 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 43 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 47 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 53 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 59 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 61 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 67 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 71 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 73 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 79 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 83 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 89 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 97 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
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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
−6.54028498103758627248048803055, −6.19887662103047928379775510811, −6.12287104031861356684574126113, −6.07327522565378431719711089833, −5.61826948104918245409595034329, −5.22892413281692106010842298341, −5.18718273278921929125291003725, −5.00928971691447017659427454585, −4.60738945026192425036822318059, −4.48534144409304495532561113064, −4.31713266192512719762584494943, −4.25542711150988847422608806160, −3.80445639997988247777380047914, −3.49493740267153850486265591929, −3.26674913046357037078758880757, −3.26276688704870715559109135092, −3.25276429686463975462586033855, −2.83608918439457315916956941846, −2.51894197072541826364369296703, −2.34816443316003368905267833315, −2.01638165915840321465968498349, −1.67458870604117024325901839506, −1.16743459697400340898597802469, −0.984865134317719784856070316227, −0.49530624874120500667433297763,
0.49530624874120500667433297763, 0.984865134317719784856070316227, 1.16743459697400340898597802469, 1.67458870604117024325901839506, 2.01638165915840321465968498349, 2.34816443316003368905267833315, 2.51894197072541826364369296703, 2.83608918439457315916956941846, 3.25276429686463975462586033855, 3.26276688704870715559109135092, 3.26674913046357037078758880757, 3.49493740267153850486265591929, 3.80445639997988247777380047914, 4.25542711150988847422608806160, 4.31713266192512719762584494943, 4.48534144409304495532561113064, 4.60738945026192425036822318059, 5.00928971691447017659427454585, 5.18718273278921929125291003725, 5.22892413281692106010842298341, 5.61826948104918245409595034329, 6.07327522565378431719711089833, 6.12287104031861356684574126113, 6.19887662103047928379775510811, 6.54028498103758627248048803055
