| L(s) = 1 | + 4·5-s + 7-s − 3·9-s − 12·17-s + 2·25-s + 4·35-s − 4·37-s + 4·41-s + 8·43-s − 12·45-s + 16·47-s + 49-s − 3·63-s + 8·67-s − 32·79-s + 9·81-s − 16·83-s − 48·85-s − 12·89-s + 4·101-s − 20·109-s − 12·119-s − 6·121-s − 28·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 0.377·7-s − 9-s − 2.91·17-s + 2/5·25-s + 0.676·35-s − 0.657·37-s + 0.624·41-s + 1.21·43-s − 1.78·45-s + 2.33·47-s + 1/7·49-s − 0.377·63-s + 0.977·67-s − 3.60·79-s + 81-s − 1.75·83-s − 5.20·85-s − 1.27·89-s + 0.398·101-s − 1.91·109-s − 1.10·119-s − 0.545·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−8.257254837686236479707568177745, −7.48402841893068646023884414833, −7.02621174890446736965278831122, −6.67949194679031572918581545842, −6.02280276296176526986201378935, −5.82613210142107824134972732669, −5.51675247245265629993912731057, −4.88671808931178581792400546928, −4.18465932715971087267449515138, −4.05428052119787997110157361479, −2.83043555474359072971239832626, −2.43940180539462742301847104796, −2.14556791802447766709825303506, −1.37997143341888120500168617355, 0,
1.37997143341888120500168617355, 2.14556791802447766709825303506, 2.43940180539462742301847104796, 2.83043555474359072971239832626, 4.05428052119787997110157361479, 4.18465932715971087267449515138, 4.88671808931178581792400546928, 5.51675247245265629993912731057, 5.82613210142107824134972732669, 6.02280276296176526986201378935, 6.67949194679031572918581545842, 7.02621174890446736965278831122, 7.48402841893068646023884414833, 8.257254837686236479707568177745
