| L(s) = 1 | − 3-s − 3·4-s − 2·7-s + 9-s + 3·12-s + 12·13-s + 5·16-s − 16·19-s + 2·21-s − 27-s + 6·28-s + 8·31-s − 3·36-s + 4·37-s − 12·39-s − 8·43-s − 5·48-s + 3·49-s − 36·52-s + 16·57-s − 4·61-s − 2·63-s − 3·64-s − 8·67-s + 4·73-s + 48·76-s + 16·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 3/2·4-s − 0.755·7-s + 1/3·9-s + 0.866·12-s + 3.32·13-s + 5/4·16-s − 3.67·19-s + 0.436·21-s − 0.192·27-s + 1.13·28-s + 1.43·31-s − 1/2·36-s + 0.657·37-s − 1.92·39-s − 1.21·43-s − 0.721·48-s + 3/7·49-s − 4.99·52-s + 2.11·57-s − 0.512·61-s − 0.251·63-s − 3/8·64-s − 0.977·67-s + 0.468·73-s + 5.50·76-s + 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 826875 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 826875 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( 1 + T \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−8.076537091159247246373578439378, −7.977292875613882940498903211299, −6.63585561999092413112257281527, −6.61969387271080601507676579543, −6.16149878863294527126110458225, −6.00194883384034511830604383024, −5.26003387597415288810047334583, −4.59872715232209164053538776872, −4.28899163544686891904773944334, −3.76479055165370209201786288014, −3.63296121870362536460359180450, −2.64765980094137076229438805768, −1.70229766620154856330402199105, −0.917037574559643692523689356640, 0,
0.917037574559643692523689356640, 1.70229766620154856330402199105, 2.64765980094137076229438805768, 3.63296121870362536460359180450, 3.76479055165370209201786288014, 4.28899163544686891904773944334, 4.59872715232209164053538776872, 5.26003387597415288810047334583, 6.00194883384034511830604383024, 6.16149878863294527126110458225, 6.61969387271080601507676579543, 6.63585561999092413112257281527, 7.977292875613882940498903211299, 8.076537091159247246373578439378
