| L(s) = 1 | − 2·3-s − 6·7-s + 9-s − 8·13-s − 2·19-s + 12·21-s − 9·25-s + 4·27-s + 16·31-s − 20·37-s + 16·39-s − 14·43-s + 13·49-s + 4·57-s − 10·61-s − 6·63-s − 30·73-s + 18·75-s − 8·79-s − 11·81-s + 48·91-s − 32·93-s + 32·97-s − 28·103-s + 24·109-s + 40·111-s − 8·117-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 2.26·7-s + 1/3·9-s − 2.21·13-s − 0.458·19-s + 2.61·21-s − 9/5·25-s + 0.769·27-s + 2.87·31-s − 3.28·37-s + 2.56·39-s − 2.13·43-s + 13/7·49-s + 0.529·57-s − 1.28·61-s − 0.755·63-s − 3.51·73-s + 2.07·75-s − 0.900·79-s − 1.22·81-s + 5.03·91-s − 3.31·93-s + 3.24·97-s − 2.75·103-s + 2.29·109-s + 3.79·111-s − 0.739·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207936 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207936 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
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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
−8.630197680642780580590235451603, −8.148422841499315186159351254036, −7.35801470679317747358663157370, −6.97775492717497319205239152750, −6.65084427660654519125997905895, −6.07596938444397512729377964410, −5.89152757780490661000300260672, −4.93646030803797952754928020473, −4.85172758929064856672456822692, −3.98454062268883454433648080079, −3.08915058635532111035563541614, −2.92269314971800745710061052414, −1.85925474650503174098776451275, 0, 0,
1.85925474650503174098776451275, 2.92269314971800745710061052414, 3.08915058635532111035563541614, 3.98454062268883454433648080079, 4.85172758929064856672456822692, 4.93646030803797952754928020473, 5.89152757780490661000300260672, 6.07596938444397512729377964410, 6.65084427660654519125997905895, 6.97775492717497319205239152750, 7.35801470679317747358663157370, 8.148422841499315186159351254036, 8.630197680642780580590235451603
