| L(s) = 1 | + 2·13-s − 2·19-s − 2·31-s + 2·43-s − 2·49-s + 2·61-s − 2·67-s + 2·73-s + 2·79-s − 2·97-s + 2·103-s + 2·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 2·13-s − 2·19-s − 2·31-s + 2·43-s − 2·49-s + 2·61-s − 2·67-s + 2·73-s + 2·79-s − 2·97-s + 2·103-s + 2·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3779136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3779136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.202009576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.202009576\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2^2$ | \( 1 + T^{4} \) |
| 59 | $C_2^2$ | \( 1 + T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + T^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
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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
−9.573243202668337204861428860891, −8.987129602501258249038015090450, −8.800539897295006893726190109579, −8.519543626831669197298280336708, −8.078500867031149834205514057112, −7.66356395948192751359859249137, −7.28209321333657416534071415815, −6.55623339616343309362604077027, −6.54547058168710754302432808774, −5.99662915432129118686496093463, −5.72350307363656571052338342490, −5.19582091798717778783100990677, −4.62147982372416173750603605302, −4.18809200065074855229515401565, −3.67935020901390904550870884872, −3.54198703495389148907683921513, −2.77944343197474956611119945110, −2.00262217531102882676359375349, −1.80286987544936973970745573275, −0.826746484795851843203846911133,
0.826746484795851843203846911133, 1.80286987544936973970745573275, 2.00262217531102882676359375349, 2.77944343197474956611119945110, 3.54198703495389148907683921513, 3.67935020901390904550870884872, 4.18809200065074855229515401565, 4.62147982372416173750603605302, 5.19582091798717778783100990677, 5.72350307363656571052338342490, 5.99662915432129118686496093463, 6.54547058168710754302432808774, 6.55623339616343309362604077027, 7.28209321333657416534071415815, 7.66356395948192751359859249137, 8.078500867031149834205514057112, 8.519543626831669197298280336708, 8.800539897295006893726190109579, 8.987129602501258249038015090450, 9.573243202668337204861428860891
