| L(s) = 1 | + 2·3-s + 2·5-s + 2·7-s + 3·9-s + 4·11-s + 2·13-s + 4·15-s + 6·17-s + 2·19-s + 4·21-s − 2·23-s + 3·25-s + 4·27-s − 6·31-s + 8·33-s + 4·35-s − 2·37-s + 4·39-s − 4·41-s + 4·43-s + 6·45-s − 18·47-s + 6·49-s + 12·51-s + 20·53-s + 8·55-s + 4·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 0.755·7-s + 9-s + 1.20·11-s + 0.554·13-s + 1.03·15-s + 1.45·17-s + 0.458·19-s + 0.872·21-s − 0.417·23-s + 3/5·25-s + 0.769·27-s − 1.07·31-s + 1.39·33-s + 0.676·35-s − 0.328·37-s + 0.640·39-s − 0.624·41-s + 0.609·43-s + 0.894·45-s − 2.62·47-s + 6/7·49-s + 1.68·51-s + 2.74·53-s + 1.07·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.945096257\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.945096257\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_4$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 54 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 58 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 18 T + 158 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 18 T + 222 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−9.265161774662914356228456599679, −8.938212050614217561674344687628, −8.554115865648414342059513737100, −8.536944800966829093560961497328, −7.75203957098456649763370360724, −7.60189826672547491517745518044, −7.03588215097266470636524751115, −6.82149031711063709499678238024, −6.15493859133749347411055667599, −5.84583346946819355992170483941, −5.21603207312641333367231469499, −5.19608158655136379746629184982, −4.17361941381657364250233413630, −4.11470428733386801546730423320, −3.43103357452412713358626233328, −3.16769132358773592150243641316, −2.46924956458657355319551876131, −1.85249753814038468668508443602, −1.49221599261218438514184861275, −0.998208230368968377456461955320,
0.998208230368968377456461955320, 1.49221599261218438514184861275, 1.85249753814038468668508443602, 2.46924956458657355319551876131, 3.16769132358773592150243641316, 3.43103357452412713358626233328, 4.11470428733386801546730423320, 4.17361941381657364250233413630, 5.19608158655136379746629184982, 5.21603207312641333367231469499, 5.84583346946819355992170483941, 6.15493859133749347411055667599, 6.82149031711063709499678238024, 7.03588215097266470636524751115, 7.60189826672547491517745518044, 7.75203957098456649763370360724, 8.536944800966829093560961497328, 8.554115865648414342059513737100, 8.938212050614217561674344687628, 9.265161774662914356228456599679
