| L(s) = 1 | + 102·3-s − 6.18e3·7-s + 3.84e3·9-s + 1.45e4·13-s + 1.60e5·19-s − 6.31e5·21-s − 5.01e5·25-s − 2.77e5·27-s + 8.71e5·31-s − 2.31e6·37-s + 1.48e6·39-s − 1.98e6·43-s + 1.71e7·49-s + 1.63e7·57-s − 3.87e7·61-s − 2.37e7·63-s + 5.60e7·67-s − 5.04e7·73-s − 5.11e7·75-s − 1.26e8·79-s − 5.34e7·81-s − 9.02e7·91-s + 8.89e7·93-s + 3.91e7·97-s + 2.26e8·103-s − 3.48e6·109-s − 2.36e8·111-s + ⋯ |
| L(s) = 1 | + 1.25·3-s − 2.57·7-s + 0.585·9-s + 0.510·13-s + 1.23·19-s − 3.24·21-s − 1.28·25-s − 0.521·27-s + 0.944·31-s − 1.23·37-s + 0.643·39-s − 0.579·43-s + 2.98·49-s + 1.55·57-s − 2.79·61-s − 1.50·63-s + 2.78·67-s − 1.77·73-s − 1.61·75-s − 3.25·79-s − 1.24·81-s − 1.31·91-s + 1.18·93-s + 0.441·97-s + 2.01·103-s − 0.0247·109-s − 1.55·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.3484757511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3484757511\) |
| \(L(5)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - 34 p T + p^{8} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 100358 p T^{2} + p^{16} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 442 p T + p^{8} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 38857702 p T^{2} + p^{16} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7294 T + p^{8} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 10482174722 T^{2} + p^{16} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 80326 T + p^{8} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 147132606722 T^{2} + p^{16} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 253546712162 T^{2} + p^{16} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 435914 T + p^{8} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 1159298 T + p^{8} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 8591852110082 T^{2} + p^{16} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 990266 T + p^{8} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2717384513282 T^{2} + p^{16} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 23095221504482 T^{2} + p^{16} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 291106443928802 T^{2} + p^{16} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 19369154 T + p^{8} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 28024294 T + p^{8} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 158683081351682 T^{2} + p^{16} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 25230142 T + p^{8} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 63401398 T + p^{8} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 2244210903661922 T^{2} + p^{16} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 1743003813196802 T^{2} + p^{16} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19550306 T + p^{8} T^{2} )^{2} \) |
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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
−11.49053414505154145963308921091, −10.56349053854162923083538484882, −9.918478414138778359359422910828, −9.835860859990394875771087756452, −9.455701263591517450389488158150, −8.678476504536478057532796303870, −8.635994496143449146200494942090, −7.68131713677518415025174312263, −7.31755255526037037205581584866, −6.75385585262704891867604965058, −5.98837169824715581834049192353, −5.97238049161267033825977352113, −4.92529367765281158042843226875, −3.98435437665883336514130857561, −3.53185361643067760964403282304, −3.10443880693455140236709943455, −2.80396224080044104416134177559, −1.92461946496145083224018077013, −1.14184090011974413464356160139, −0.12903622964390962980346917248,
0.12903622964390962980346917248, 1.14184090011974413464356160139, 1.92461946496145083224018077013, 2.80396224080044104416134177559, 3.10443880693455140236709943455, 3.53185361643067760964403282304, 3.98435437665883336514130857561, 4.92529367765281158042843226875, 5.97238049161267033825977352113, 5.98837169824715581834049192353, 6.75385585262704891867604965058, 7.31755255526037037205581584866, 7.68131713677518415025174312263, 8.635994496143449146200494942090, 8.678476504536478057532796303870, 9.455701263591517450389488158150, 9.835860859990394875771087756452, 9.918478414138778359359422910828, 10.56349053854162923083538484882, 11.49053414505154145963308921091
