| L(s) = 1 | + 3-s + 9-s + 8·23-s − 10·25-s + 27-s − 8·29-s − 16·43-s − 24·47-s − 10·49-s + 24·53-s − 8·67-s + 8·69-s − 8·71-s − 20·73-s − 10·75-s + 81-s − 8·87-s − 12·97-s − 8·101-s − 6·121-s + 127-s − 16·129-s + 131-s + 137-s + 139-s − 24·141-s − 10·147-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 1.66·23-s − 2·25-s + 0.192·27-s − 1.48·29-s − 2.43·43-s − 3.50·47-s − 1.42·49-s + 3.29·53-s − 0.977·67-s + 0.963·69-s − 0.949·71-s − 2.34·73-s − 1.15·75-s + 1/9·81-s − 0.857·87-s − 1.21·97-s − 0.796·101-s − 0.545·121-s + 0.0887·127-s − 1.40·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.02·141-s − 0.824·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−8.503024042291174545390595385787, −7.76841700352222764456930738236, −7.67554775307177371631314255080, −6.81542041238937589948552667187, −6.80910210319960057293306406499, −6.04519620219891673101591163967, −5.36858848597467135489376840463, −5.19382499916710112913058943849, −4.36074490911347519116057298154, −3.97633443717640489814889299763, −3.13420541549434036877759112013, −3.06807634409715093203408940942, −1.86962407001750988737926530376, −1.59108885184574423909298130831, 0,
1.59108885184574423909298130831, 1.86962407001750988737926530376, 3.06807634409715093203408940942, 3.13420541549434036877759112013, 3.97633443717640489814889299763, 4.36074490911347519116057298154, 5.19382499916710112913058943849, 5.36858848597467135489376840463, 6.04519620219891673101591163967, 6.80910210319960057293306406499, 6.81542041238937589948552667187, 7.67554775307177371631314255080, 7.76841700352222764456930738236, 8.503024042291174545390595385787
