| L(s) = 1 | + 20·2-s + 44·4-s − 3.68e3·8-s − 1.61e3·9-s − 9.49e3·11-s − 5.44e4·16-s − 3.22e4·18-s − 1.89e5·22-s − 1.51e5·23-s + 6.73e4·25-s − 2.18e5·29-s + 6.08e3·32-s − 7.10e4·36-s + 3.99e5·37-s − 8.38e5·43-s − 4.17e5·44-s − 3.02e6·46-s + 1.34e6·50-s − 9.33e5·53-s − 4.37e6·58-s + 9.36e6·64-s + 1.98e6·67-s − 1.99e6·71-s + 5.93e6·72-s + 7.98e6·74-s − 5.66e6·79-s − 2.17e6·81-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 0.343·4-s − 2.54·8-s − 0.737·9-s − 2.15·11-s − 3.32·16-s − 1.30·18-s − 3.80·22-s − 2.58·23-s + 0.861·25-s − 1.66·29-s + 0.0328·32-s − 0.253·36-s + 1.29·37-s − 1.60·43-s − 0.739·44-s − 4.57·46-s + 1.52·50-s − 0.861·53-s − 2.94·58-s + 4.46·64-s + 0.807·67-s − 0.662·71-s + 1.87·72-s + 2.29·74-s − 1.29·79-s − 0.455·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - 5 p T + p^{7} T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 + 538 p T^{2} + p^{14} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 13462 p T^{2} + p^{14} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4748 T + p^{7} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 114542594 T^{2} + p^{14} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 403537214 T^{2} + p^{14} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 1111474478 T^{2} + p^{14} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 75520 T + p^{7} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 109366 T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 55002872222 T^{2} + p^{14} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 199650 T + p^{7} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 143033647762 T^{2} + p^{14} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 419340 T + p^{7} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 436334014 p^{2} T^{2} + p^{14} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 8810 p T + p^{7} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 4881390140638 T^{2} + p^{14} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6255701891042 T^{2} + p^{14} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 994180 T + p^{7} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 998912 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 17080414200434 T^{2} + p^{14} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 2832904 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 29962382079506 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 55405836255058 T^{2} + p^{14} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 49002058500386 T^{2} + p^{14} T^{4} \) |
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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
−13.70933344659560221343222271568, −13.31365027869678141555567101308, −12.80799179250529128619074953966, −12.49627439262702494438365395102, −11.65463889494968584777684781769, −11.24397969993446878655476785682, −10.17115436669025475368770816928, −9.768610021307935429377767614715, −8.840094157388688516184754939445, −8.235905049785791330993917505803, −7.66712086006085880323665654767, −6.30142052190697300573041861030, −5.74594440579686685753994639545, −5.28210182733921642633841235534, −4.62072186732419298821106989452, −3.80670749923903152465259144290, −3.02850999339261372677103328532, −2.31788556249958549691827469901, 0, 0,
2.31788556249958549691827469901, 3.02850999339261372677103328532, 3.80670749923903152465259144290, 4.62072186732419298821106989452, 5.28210182733921642633841235534, 5.74594440579686685753994639545, 6.30142052190697300573041861030, 7.66712086006085880323665654767, 8.235905049785791330993917505803, 8.840094157388688516184754939445, 9.768610021307935429377767614715, 10.17115436669025475368770816928, 11.24397969993446878655476785682, 11.65463889494968584777684781769, 12.49627439262702494438365395102, 12.80799179250529128619074953966, 13.31365027869678141555567101308, 13.70933344659560221343222271568
