| L(s) = 1 | + 2·3-s − 2·5-s − 2·7-s + 4·11-s − 4·15-s + 8·19-s − 4·21-s − 2·23-s + 3·25-s − 2·27-s + 4·31-s + 8·33-s + 4·35-s − 4·37-s + 4·41-s + 14·43-s − 10·47-s − 8·49-s + 16·53-s − 8·55-s + 16·57-s + 8·59-s + 4·61-s + 18·67-s − 4·69-s + 4·71-s − 8·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 0.755·7-s + 1.20·11-s − 1.03·15-s + 1.83·19-s − 0.872·21-s − 0.417·23-s + 3/5·25-s − 0.384·27-s + 0.718·31-s + 1.39·33-s + 0.676·35-s − 0.657·37-s + 0.624·41-s + 2.13·43-s − 1.45·47-s − 8/7·49-s + 2.19·53-s − 1.07·55-s + 2.11·57-s + 1.04·59-s + 0.512·61-s + 2.19·67-s − 0.481·69-s + 0.474·71-s − 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.697250844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.697250844\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 14 T + 132 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 10 T + 116 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 16 T + 158 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 122 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 18 T + 212 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 134 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 172 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−9.543729856134368219682187800953, −9.485390641777359016768945726897, −9.106596460467922606661847011570, −8.672714553394289769313921727188, −8.174740303887078720907425032421, −7.994308452090982718640075283941, −7.54198666089941367045483251094, −7.04798069647412296691920975021, −6.55990382620793343410694020875, −6.46542553214848439283986566875, −5.46888555848031269390399544413, −5.42384512169383981413181618519, −4.63190343746558618125675085263, −3.94707571883818012089443160324, −3.72168676285837127845647845823, −3.41127724929545797944794431389, −2.65589433541991244572373284709, −2.50478899943272120890314001504, −1.37191962178184502383989300481, −0.69756775374337419424295897806,
0.69756775374337419424295897806, 1.37191962178184502383989300481, 2.50478899943272120890314001504, 2.65589433541991244572373284709, 3.41127724929545797944794431389, 3.72168676285837127845647845823, 3.94707571883818012089443160324, 4.63190343746558618125675085263, 5.42384512169383981413181618519, 5.46888555848031269390399544413, 6.46542553214848439283986566875, 6.55990382620793343410694020875, 7.04798069647412296691920975021, 7.54198666089941367045483251094, 7.994308452090982718640075283941, 8.174740303887078720907425032421, 8.672714553394289769313921727188, 9.106596460467922606661847011570, 9.485390641777359016768945726897, 9.543729856134368219682187800953
