| L(s) = 1 | − 4·4-s − 7-s − 5·9-s − 3·11-s − 10·13-s + 12·16-s − 6·17-s − 4·19-s − 12·23-s + 25-s + 4·28-s + 20·36-s + 4·37-s + 24·41-s + 12·44-s + 49-s + 40·52-s + 24·53-s − 16·61-s + 5·63-s − 32·64-s − 8·67-s + 24·68-s − 4·73-s + 16·76-s + 3·77-s + 16·81-s + ⋯ |
| L(s) = 1 | − 2·4-s − 0.377·7-s − 5/3·9-s − 0.904·11-s − 2.77·13-s + 3·16-s − 1.45·17-s − 0.917·19-s − 2.50·23-s + 1/5·25-s + 0.755·28-s + 10/3·36-s + 0.657·37-s + 3.74·41-s + 1.80·44-s + 1/7·49-s + 5.54·52-s + 3.29·53-s − 2.04·61-s + 0.629·63-s − 4·64-s − 0.977·67-s + 2.91·68-s − 0.468·73-s + 1.83·76-s + 0.341·77-s + 16/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1037575 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1037575 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( 1 + T \) |
| 11 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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
−7.80156257853413862136981277007, −7.67950449335459092353880992506, −7.28726628142540726038141332638, −6.30078072165756181986797640838, −5.98086532326102961421297628547, −5.49671842174989089979243313601, −5.30121167828360660438204787375, −4.48390865811726709341858080699, −4.26526382517520030491171627830, −4.09257990991624762122042789987, −2.90308901018164947604047652054, −2.61198254943438177491615875407, −2.20945739428189555232801439935, −0.42426621729141121803731134362, 0,
0.42426621729141121803731134362, 2.20945739428189555232801439935, 2.61198254943438177491615875407, 2.90308901018164947604047652054, 4.09257990991624762122042789987, 4.26526382517520030491171627830, 4.48390865811726709341858080699, 5.30121167828360660438204787375, 5.49671842174989089979243313601, 5.98086532326102961421297628547, 6.30078072165756181986797640838, 7.28726628142540726038141332638, 7.67950449335459092353880992506, 7.80156257853413862136981277007
