| L(s) = 1 | − 4·5-s + 6·13-s + 4·17-s + 7·19-s + 4·23-s + 5·25-s + 16·29-s − 5·31-s − 3·37-s + 16·41-s + 22·43-s − 4·47-s + 4·53-s − 12·59-s − 2·61-s − 24·65-s + 3·67-s − 24·71-s + 73-s − 79-s + 24·83-s − 16·85-s − 8·89-s − 28·95-s + 4·97-s + 3·103-s − 12·107-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1.66·13-s + 0.970·17-s + 1.60·19-s + 0.834·23-s + 25-s + 2.97·29-s − 0.898·31-s − 0.493·37-s + 2.49·41-s + 3.35·43-s − 0.583·47-s + 0.549·53-s − 1.56·59-s − 0.256·61-s − 2.97·65-s + 0.366·67-s − 2.84·71-s + 0.117·73-s − 0.112·79-s + 2.63·83-s − 1.73·85-s − 0.847·89-s − 2.87·95-s + 0.406·97-s + 0.295·103-s − 1.16·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.888659053\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.888659053\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 4 T - 31 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 8 T - 25 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.559561959863017829709410804226, −8.529369687634611568325823557103, −7.78087061403301152720187519574, −7.72223129594299051355931446332, −7.32209167454931939056938177301, −7.26360145984578740217757323890, −6.46938486397854112002955414682, −5.98531443713491862603752124567, −5.97253008847126941210587264287, −5.39336738003913672917276657487, −4.69248046830280884918663871053, −4.62288230374164225907320541892, −3.91336809577942005458918695130, −3.86966211393064922194364254045, −3.10624492650664100747341317179, −3.10590490004609489005453155726, −2.50530360609800752838603368437, −1.47193992646419057075097411795, −0.857190845935932183204033553027, −0.77852974664758079529937657563,
0.77852974664758079529937657563, 0.857190845935932183204033553027, 1.47193992646419057075097411795, 2.50530360609800752838603368437, 3.10590490004609489005453155726, 3.10624492650664100747341317179, 3.86966211393064922194364254045, 3.91336809577942005458918695130, 4.62288230374164225907320541892, 4.69248046830280884918663871053, 5.39336738003913672917276657487, 5.97253008847126941210587264287, 5.98531443713491862603752124567, 6.46938486397854112002955414682, 7.26360145984578740217757323890, 7.32209167454931939056938177301, 7.72223129594299051355931446332, 7.78087061403301152720187519574, 8.529369687634611568325823557103, 8.559561959863017829709410804226
