L(s) = 1 | − 2·3-s − 2·4-s − 7-s − 3·9-s + 4·12-s + 4·16-s − 4·19-s + 2·21-s + 25-s + 14·27-s + 2·28-s + 6·29-s + 8·31-s + 6·36-s + 4·37-s − 18·47-s − 8·48-s + 49-s + 24·53-s + 8·57-s + 3·63-s − 8·64-s − 2·75-s + 8·76-s − 4·81-s − 24·83-s − 4·84-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 4-s − 0.377·7-s − 9-s + 1.15·12-s + 16-s − 0.917·19-s + 0.436·21-s + 1/5·25-s + 2.69·27-s + 0.377·28-s + 1.11·29-s + 1.43·31-s + 36-s + 0.657·37-s − 2.62·47-s − 1.15·48-s + 1/7·49-s + 3.29·53-s + 1.05·57-s + 0.377·63-s − 64-s − 0.230·75-s + 0.917·76-s − 4/9·81-s − 2.63·83-s − 0.436·84-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 137200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( 1 + T \) |
good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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
−8.829468246037740149648448974500, −8.722193961422815471879095064652, −8.322081181544343689933636986956, −7.80156257853413862136981277007, −6.91010810779795429087322035266, −6.30078072165756181986797640838, −6.26051630300961999746999469622, −5.30121167828360660438204787375, −5.29742546009269090365766846421, −4.48390865811726709341858080699, −4.02723544557718882488682513095, −3.04856603410542935889592969685, −2.61198254943438177491615875407, −1.00806560019316329290903940266, 0,
1.00806560019316329290903940266, 2.61198254943438177491615875407, 3.04856603410542935889592969685, 4.02723544557718882488682513095, 4.48390865811726709341858080699, 5.29742546009269090365766846421, 5.30121167828360660438204787375, 6.26051630300961999746999469622, 6.30078072165756181986797640838, 6.91010810779795429087322035266, 7.80156257853413862136981277007, 8.322081181544343689933636986956, 8.722193961422815471879095064652, 8.829468246037740149648448974500