| L(s) = 1 | − 3-s + 9-s − 8·11-s − 4·13-s − 4·17-s − 8·19-s − 6·25-s − 27-s − 12·29-s + 8·33-s + 4·39-s + 12·41-s − 7·49-s + 4·51-s + 4·53-s + 8·57-s − 4·61-s + 6·75-s − 16·79-s + 81-s + 12·87-s + 12·89-s − 8·99-s + 24·107-s − 4·117-s + 26·121-s − 12·123-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 2.41·11-s − 1.10·13-s − 0.970·17-s − 1.83·19-s − 6/5·25-s − 0.192·27-s − 2.22·29-s + 1.39·33-s + 0.640·39-s + 1.87·41-s − 49-s + 0.560·51-s + 0.549·53-s + 1.05·57-s − 0.512·61-s + 0.692·75-s − 1.80·79-s + 1/9·81-s + 1.28·87-s + 1.27·89-s − 0.804·99-s + 2.32·107-s − 0.369·117-s + 2.36·121-s − 1.08·123-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 677376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 677376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.62403207119791885504595168937, −7.57346950816982656575487446761, −7.22685430203741355342227556898, −6.28266974817041814701199375242, −6.14969218484336320739052245813, −5.59620622280165428169903914653, −5.00670845718665013316471514453, −4.83326855247570334999188527992, −4.12187187931469968184698878478, −3.71533601983866002659839530844, −2.51659494204614894722398853037, −2.50123083621432021396021244496, −1.80970977960128750436372604769, 0, 0,
1.80970977960128750436372604769, 2.50123083621432021396021244496, 2.51659494204614894722398853037, 3.71533601983866002659839530844, 4.12187187931469968184698878478, 4.83326855247570334999188527992, 5.00670845718665013316471514453, 5.59620622280165428169903914653, 6.14969218484336320739052245813, 6.28266974817041814701199375242, 7.22685430203741355342227556898, 7.57346950816982656575487446761, 7.62403207119791885504595168937
