| L(s) = 1 | − 2·3-s − 6·7-s + 2·9-s + 6·13-s − 2·17-s + 8·19-s + 12·21-s − 2·23-s − 6·27-s − 2·37-s − 12·39-s − 20·41-s − 10·43-s − 6·47-s + 18·49-s + 4·51-s − 10·53-s − 16·57-s − 24·59-s − 4·61-s − 12·63-s + 2·67-s + 4·69-s − 2·73-s − 16·79-s + 11·81-s − 10·83-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 2.26·7-s + 2/3·9-s + 1.66·13-s − 0.485·17-s + 1.83·19-s + 2.61·21-s − 0.417·23-s − 1.15·27-s − 0.328·37-s − 1.92·39-s − 3.12·41-s − 1.52·43-s − 0.875·47-s + 18/7·49-s + 0.560·51-s − 1.37·53-s − 2.11·57-s − 3.12·59-s − 0.512·61-s − 1.51·63-s + 0.244·67-s + 0.481·69-s − 0.234·73-s − 1.80·79-s + 11/9·81-s − 1.09·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 138 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−9.492720387548640946458888795814, −8.912927045451480904730248988512, −8.283539223919954088699060381263, −8.238087443664100648297461376930, −7.35466130558282302581545567147, −7.08197431740022071042206041486, −6.60444173453214833013871216286, −6.40693536831449581400013557028, −5.89290166053203571505447211701, −5.81728526750977657581662811718, −5.14296477111565930907404971565, −4.76596156559381746265679850396, −4.08010266609314751459190029639, −3.43677035799624185483612676385, −3.26524033774252583616087916396, −3.00867318863603356950809490039, −1.68082192265665586254738887024, −1.38134933259869946030630000976, 0, 0,
1.38134933259869946030630000976, 1.68082192265665586254738887024, 3.00867318863603356950809490039, 3.26524033774252583616087916396, 3.43677035799624185483612676385, 4.08010266609314751459190029639, 4.76596156559381746265679850396, 5.14296477111565930907404971565, 5.81728526750977657581662811718, 5.89290166053203571505447211701, 6.40693536831449581400013557028, 6.60444173453214833013871216286, 7.08197431740022071042206041486, 7.35466130558282302581545567147, 8.238087443664100648297461376930, 8.283539223919954088699060381263, 8.912927045451480904730248988512, 9.492720387548640946458888795814
