| L(s) = 1 | − 3-s + 9-s + 12·11-s + 4·13-s + 12·23-s − 10·25-s − 27-s − 12·33-s + 4·37-s − 4·39-s − 24·47-s + 49-s − 20·61-s − 12·69-s − 12·71-s − 20·73-s + 10·75-s + 81-s + 24·83-s − 20·97-s + 12·99-s + 12·107-s + 28·109-s − 4·111-s + 4·117-s + 86·121-s + 127-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 3.61·11-s + 1.10·13-s + 2.50·23-s − 2·25-s − 0.192·27-s − 2.08·33-s + 0.657·37-s − 0.640·39-s − 3.50·47-s + 1/7·49-s − 2.56·61-s − 1.44·69-s − 1.42·71-s − 2.34·73-s + 1.15·75-s + 1/9·81-s + 2.63·83-s − 2.03·97-s + 1.20·99-s + 1.16·107-s + 2.68·109-s − 0.379·111-s + 0.369·117-s + 7.81·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84672 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84672 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.648773141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.648773141\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 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
−9.542934180818189537522408595040, −9.072046417107109896038765861128, −9.052436712736537682628040663330, −8.300043091642042619777544931378, −7.58881932943940700511157248554, −6.93446262615281462668117301613, −6.58911559432542935222512700270, −6.11056271857714724112691707067, −5.84091064569873502208467244139, −4.63917458628061830491997962672, −4.46733872516772697970621091271, −3.53462971911322619523210724383, −3.36010318341016445358722759890, −1.58867103529813382887688476545, −1.28052617279627540057290389597,
1.28052617279627540057290389597, 1.58867103529813382887688476545, 3.36010318341016445358722759890, 3.53462971911322619523210724383, 4.46733872516772697970621091271, 4.63917458628061830491997962672, 5.84091064569873502208467244139, 6.11056271857714724112691707067, 6.58911559432542935222512700270, 6.93446262615281462668117301613, 7.58881932943940700511157248554, 8.300043091642042619777544931378, 9.052436712736537682628040663330, 9.072046417107109896038765861128, 9.542934180818189537522408595040
