| L(s) = 1 | − 2·3-s + 2·5-s + 3·9-s − 4·13-s − 4·15-s − 25-s − 4·27-s − 8·31-s − 4·37-s + 8·39-s + 4·41-s − 8·43-s + 6·45-s + 2·49-s + 20·53-s − 8·65-s − 8·67-s + 32·71-s + 2·75-s − 8·79-s + 5·81-s − 24·83-s + 20·89-s + 16·93-s + 8·107-s + 8·111-s − 12·117-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 9-s − 1.10·13-s − 1.03·15-s − 1/5·25-s − 0.769·27-s − 1.43·31-s − 0.657·37-s + 1.28·39-s + 0.624·41-s − 1.21·43-s + 0.894·45-s + 2/7·49-s + 2.74·53-s − 0.992·65-s − 0.977·67-s + 3.79·71-s + 0.230·75-s − 0.900·79-s + 5/9·81-s − 2.63·83-s + 2.11·89-s + 1.65·93-s + 0.773·107-s + 0.759·111-s − 1.10·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.391046533691483698004382219288, −7.72566049296232208126304466870, −7.17212008872630953090710242182, −7.05194887310095876236379718088, −6.42608587339790521684252786803, −5.88581502075612967461940245349, −5.56983356275681900785379754785, −5.09978459961077278060048007839, −4.77602747706575131165960838209, −3.95774934920823868374611495467, −3.54312511922683791909994900456, −2.44692008831488081083966798293, −2.09803110754033007053716721218, −1.18595912589171287323913243070, 0,
1.18595912589171287323913243070, 2.09803110754033007053716721218, 2.44692008831488081083966798293, 3.54312511922683791909994900456, 3.95774934920823868374611495467, 4.77602747706575131165960838209, 5.09978459961077278060048007839, 5.56983356275681900785379754785, 5.88581502075612967461940245349, 6.42608587339790521684252786803, 7.05194887310095876236379718088, 7.17212008872630953090710242182, 7.72566049296232208126304466870, 8.391046533691483698004382219288
