L(s) = 1 | + 3-s − 4·4-s + 6·5-s − 4·7-s + 9-s − 4·12-s + 6·15-s + 12·16-s − 2·17-s − 24·20-s − 4·21-s + 17·25-s + 27-s + 16·28-s − 24·35-s − 4·36-s − 8·37-s − 6·41-s − 14·43-s + 6·45-s − 12·47-s + 12·48-s + 9·49-s − 2·51-s + 12·59-s − 24·60-s − 4·63-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 2·4-s + 2.68·5-s − 1.51·7-s + 1/3·9-s − 1.15·12-s + 1.54·15-s + 3·16-s − 0.485·17-s − 5.36·20-s − 0.872·21-s + 17/5·25-s + 0.192·27-s + 3.02·28-s − 4.05·35-s − 2/3·36-s − 1.31·37-s − 0.937·41-s − 2.13·43-s + 0.894·45-s − 1.75·47-s + 1.73·48-s + 9/7·49-s − 0.280·51-s + 1.56·59-s − 3.09·60-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 382347 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 382347 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−8.618966621189813626434174913816, −8.469847884947395544150995937007, −7.58523751895654898654671253270, −6.71025591249190433133409575281, −6.66463575826783700250676263362, −5.91938342687192026845077841351, −5.62865745280264567981666866924, −5.14306131647545403565477997005, −4.71647271720364633813685518770, −3.96826933720343816290629867972, −3.27969494379582339775711456114, −3.00632648437536992753290178800, −1.98195814340014591526633802801, −1.44650813195969740431651518332, 0,
1.44650813195969740431651518332, 1.98195814340014591526633802801, 3.00632648437536992753290178800, 3.27969494379582339775711456114, 3.96826933720343816290629867972, 4.71647271720364633813685518770, 5.14306131647545403565477997005, 5.62865745280264567981666866924, 5.91938342687192026845077841351, 6.66463575826783700250676263362, 6.71025591249190433133409575281, 7.58523751895654898654671253270, 8.469847884947395544150995937007, 8.618966621189813626434174913816