| L(s) = 1 | + 3-s + 2·7-s + 9-s + 4·13-s − 8·19-s + 2·21-s − 10·25-s + 27-s + 16·31-s + 4·37-s + 4·39-s − 8·43-s + 3·49-s − 8·57-s − 20·61-s + 2·63-s + 16·67-s − 20·73-s − 10·75-s − 8·79-s + 81-s + 8·91-s + 16·93-s − 20·97-s + 16·103-s + 28·109-s + 4·111-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s + 1.10·13-s − 1.83·19-s + 0.436·21-s − 2·25-s + 0.192·27-s + 2.87·31-s + 0.657·37-s + 0.640·39-s − 1.21·43-s + 3/7·49-s − 1.05·57-s − 2.56·61-s + 0.251·63-s + 1.95·67-s − 2.34·73-s − 1.15·75-s − 0.900·79-s + 1/9·81-s + 0.838·91-s + 1.65·93-s − 2.03·97-s + 1.57·103-s + 2.68·109-s + 0.379·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21168 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21168 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.511096278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.511096278\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−10.85008427837924164556556041776, −10.10483686416197845980665091731, −9.963412816934859098438744607306, −9.072046417107109896038765861128, −8.418500928498397092746321719715, −8.300043091642042619777544931378, −7.74489471466623660805940258686, −6.93446262615281462668117301613, −6.11056271857714724112691707067, −5.98540521798352690518559815387, −4.63917458628061830491997962672, −4.39050801784138122129668187119, −3.53462971911322619523210724383, −2.51752502799515976298852414682, −1.58867103529813382887688476545,
1.58867103529813382887688476545, 2.51752502799515976298852414682, 3.53462971911322619523210724383, 4.39050801784138122129668187119, 4.63917458628061830491997962672, 5.98540521798352690518559815387, 6.11056271857714724112691707067, 6.93446262615281462668117301613, 7.74489471466623660805940258686, 8.300043091642042619777544931378, 8.418500928498397092746321719715, 9.072046417107109896038765861128, 9.963412816934859098438744607306, 10.10483686416197845980665091731, 10.85008427837924164556556041776
