L(s) = 1 | + 3-s + 4·7-s + 9-s − 12·13-s + 4·21-s − 10·25-s + 27-s + 20·31-s − 4·37-s − 12·39-s − 16·43-s − 2·49-s − 4·61-s + 4·63-s − 8·67-s − 20·73-s − 10·75-s − 12·79-s + 81-s − 48·91-s + 20·93-s − 12·97-s − 20·103-s + 4·109-s − 4·111-s − 12·117-s − 6·121-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.51·7-s + 1/3·9-s − 3.32·13-s + 0.872·21-s − 2·25-s + 0.192·27-s + 3.59·31-s − 0.657·37-s − 1.92·39-s − 2.43·43-s − 2/7·49-s − 0.512·61-s + 0.503·63-s − 0.977·67-s − 2.34·73-s − 1.15·75-s − 1.35·79-s + 1/9·81-s − 5.03·91-s + 2.07·93-s − 1.21·97-s − 1.97·103-s + 0.383·109-s − 0.379·111-s − 1.10·117-s − 0.545·121-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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−8.352864246059164846839031639199, −7.86636283231227873209414758068, −7.67554775307177371631314255080, −7.00186001179257888349709337610, −6.80910210319960057293306406499, −5.96427857852485389909948133454, −5.36858848597467135489376840463, −4.86024035915155668889978626292, −4.46736447254899856037450817253, −4.36074490911347519116057298154, −3.06807634409715093203408940942, −2.79778517840079752431830788770, −2.02071693225675944817521065685, −1.59108885184574423909298130831, 0,
1.59108885184574423909298130831, 2.02071693225675944817521065685, 2.79778517840079752431830788770, 3.06807634409715093203408940942, 4.36074490911347519116057298154, 4.46736447254899856037450817253, 4.86024035915155668889978626292, 5.36858848597467135489376840463, 5.96427857852485389909948133454, 6.80910210319960057293306406499, 7.00186001179257888349709337610, 7.67554775307177371631314255080, 7.86636283231227873209414758068, 8.352864246059164846839031639199