| L(s) = 1 | + 3-s + 4-s + 10·7-s + 9-s + 12-s − 8·13-s + 16-s − 2·19-s + 10·21-s − 25-s + 27-s + 10·28-s − 8·31-s + 36-s − 2·37-s − 8·39-s − 14·43-s + 48-s + 61·49-s − 8·52-s − 2·57-s − 20·61-s + 10·63-s + 64-s − 8·67-s + 4·73-s − 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s + 3.77·7-s + 1/3·9-s + 0.288·12-s − 2.21·13-s + 1/4·16-s − 0.458·19-s + 2.18·21-s − 1/5·25-s + 0.192·27-s + 1.88·28-s − 1.43·31-s + 1/6·36-s − 0.328·37-s − 1.28·39-s − 2.13·43-s + 0.144·48-s + 61/7·49-s − 1.10·52-s − 0.264·57-s − 2.56·61-s + 1.25·63-s + 1/8·64-s − 0.977·67-s + 0.468·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90828 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90828 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.833374908\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.833374908\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 - T \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−9.487370424405800916973935371322, −9.109497480949641538548778552338, −8.443424828695416122862075486969, −7.987480521096063799248885737020, −7.83268146353528315500962133734, −7.30170311636574927756275463221, −6.97231192698558333316844758012, −5.87958673389096713663595768150, −5.13801159009478652046400048130, −4.85512283254279582558561488230, −4.61358895125554701935773223253, −3.70435541571971822099748167322, −2.54786728707166221497736882776, −1.92188031498074361434341547863, −1.63272357936029339660940403376,
1.63272357936029339660940403376, 1.92188031498074361434341547863, 2.54786728707166221497736882776, 3.70435541571971822099748167322, 4.61358895125554701935773223253, 4.85512283254279582558561488230, 5.13801159009478652046400048130, 5.87958673389096713663595768150, 6.97231192698558333316844758012, 7.30170311636574927756275463221, 7.83268146353528315500962133734, 7.987480521096063799248885737020, 8.443424828695416122862075486969, 9.109497480949641538548778552338, 9.487370424405800916973935371322
