L(s) = 1 | + 3-s + 6·5-s + 7-s + 6·11-s + 6·15-s − 12·17-s + 7·19-s + 21-s + 19·25-s − 27-s − 5·31-s + 6·33-s + 6·35-s + 37-s + 6·47-s − 6·49-s − 12·51-s + 36·55-s + 7·57-s − 3·67-s + 15·73-s + 19·75-s + 6·77-s − 27·79-s − 81-s + 12·83-s − 72·85-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 2.68·5-s + 0.377·7-s + 1.80·11-s + 1.54·15-s − 2.91·17-s + 1.60·19-s + 0.218·21-s + 19/5·25-s − 0.192·27-s − 0.898·31-s + 1.04·33-s + 1.01·35-s + 0.164·37-s + 0.875·47-s − 6/7·49-s − 1.68·51-s + 4.85·55-s + 0.927·57-s − 0.366·67-s + 1.75·73-s + 2.19·75-s + 0.683·77-s − 3.03·79-s − 1/9·81-s + 1.31·83-s − 7.80·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.801564085\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.801564085\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 15 T + 148 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 27 T + 322 T^{2} + 27 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 137 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−9.537100351246817620864385934229, −9.502575621615633727792216860664, −9.088715317658547562817399655190, −8.711801670108125335304987782737, −8.631588017897224189956207000244, −7.78469038409165189007121411511, −7.13177564038868560636796121004, −6.94866409447265996098729505450, −6.43165242562004236621811930512, −6.12087609622951812353079906263, −5.80905255413100459293293919254, −5.24621234750541609240667861761, −4.77742325447730044326959309581, −4.35739821575581606025693989676, −3.72158531733819014535825401804, −3.10545492636266755888175225462, −2.45977269823779518765667089920, −1.92350600600006433473941044870, −1.80501360781606946745887743509, −1.00834241934176388396059998862,
1.00834241934176388396059998862, 1.80501360781606946745887743509, 1.92350600600006433473941044870, 2.45977269823779518765667089920, 3.10545492636266755888175225462, 3.72158531733819014535825401804, 4.35739821575581606025693989676, 4.77742325447730044326959309581, 5.24621234750541609240667861761, 5.80905255413100459293293919254, 6.12087609622951812353079906263, 6.43165242562004236621811930512, 6.94866409447265996098729505450, 7.13177564038868560636796121004, 7.78469038409165189007121411511, 8.631588017897224189956207000244, 8.711801670108125335304987782737, 9.088715317658547562817399655190, 9.502575621615633727792216860664, 9.537100351246817620864385934229