| L(s) = 1 | − 2·3-s + 3·9-s − 8·11-s − 4·17-s − 8·19-s − 2·25-s − 4·27-s + 16·33-s − 20·41-s − 24·43-s − 6·49-s + 8·51-s + 16·57-s + 8·59-s − 8·67-s + 4·73-s + 4·75-s + 5·81-s + 8·83-s − 12·89-s + 28·97-s − 24·99-s − 24·107-s + 4·113-s + 26·121-s + 40·123-s + 127-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s − 2.41·11-s − 0.970·17-s − 1.83·19-s − 2/5·25-s − 0.769·27-s + 2.78·33-s − 3.12·41-s − 3.65·43-s − 6/7·49-s + 1.12·51-s + 2.11·57-s + 1.04·59-s − 0.977·67-s + 0.468·73-s + 0.461·75-s + 5/9·81-s + 0.878·83-s − 1.27·89-s + 2.84·97-s − 2.41·99-s − 2.32·107-s + 0.376·113-s + 2.36·121-s + 3.60·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−10.28998804658677757680466577619, −10.07245506107097851006983775266, −9.365635563042269871264855048440, −8.636218910639623380462350540490, −8.230870359582235551156994314656, −8.214583816897408615776573430204, −7.45216473764220302007278354195, −6.91079549811893204270195542090, −6.46934467573382852912407027576, −6.35770759782183590956851336511, −5.35749205002680040951818323440, −5.30775673772913385750429218890, −4.79742178821599754690515469672, −4.41215189075674565850640628993, −3.56423687591730804419130023020, −3.02446976059117841138850052879, −2.10278277956094620876076270023, −1.80842388564724944439291765894, 0, 0,
1.80842388564724944439291765894, 2.10278277956094620876076270023, 3.02446976059117841138850052879, 3.56423687591730804419130023020, 4.41215189075674565850640628993, 4.79742178821599754690515469672, 5.30775673772913385750429218890, 5.35749205002680040951818323440, 6.35770759782183590956851336511, 6.46934467573382852912407027576, 6.91079549811893204270195542090, 7.45216473764220302007278354195, 8.214583816897408615776573430204, 8.230870359582235551156994314656, 8.636218910639623380462350540490, 9.365635563042269871264855048440, 10.07245506107097851006983775266, 10.28998804658677757680466577619
