| L(s) = 1 | − 3-s − 4·4-s − 4·7-s + 9-s + 4·12-s + 6·13-s + 12·16-s + 4·19-s + 4·21-s − 6·25-s − 27-s + 16·28-s − 10·31-s − 4·36-s + 16·37-s − 6·39-s − 2·43-s − 12·48-s − 2·49-s − 24·52-s − 4·57-s − 16·61-s − 4·63-s − 32·64-s − 30·67-s + 24·73-s + 6·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2·4-s − 1.51·7-s + 1/3·9-s + 1.15·12-s + 1.66·13-s + 3·16-s + 0.917·19-s + 0.872·21-s − 6/5·25-s − 0.192·27-s + 3.02·28-s − 1.79·31-s − 2/3·36-s + 2.63·37-s − 0.960·39-s − 0.304·43-s − 1.73·48-s − 2/7·49-s − 3.32·52-s − 0.529·57-s − 2.04·61-s − 0.503·63-s − 4·64-s − 3.66·67-s + 2.80·73-s + 0.692·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49923 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49923 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 43 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
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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
−9.761758139217697478982108964917, −9.360930401498180975205138352635, −9.000644041338349206828923104341, −8.510537086333408831410494114245, −7.64365379837943969382466665403, −7.51672561268763905855633948060, −6.27273821245987508595068296869, −5.96338923144874103909303632728, −5.71816265250030609184960935219, −4.74387622427760402111791858890, −4.27563665262617109733543688482, −3.47776319336265678852062283368, −3.29623080699256347856604127938, −1.27597065440948837170642206895, 0,
1.27597065440948837170642206895, 3.29623080699256347856604127938, 3.47776319336265678852062283368, 4.27563665262617109733543688482, 4.74387622427760402111791858890, 5.71816265250030609184960935219, 5.96338923144874103909303632728, 6.27273821245987508595068296869, 7.51672561268763905855633948060, 7.64365379837943969382466665403, 8.510537086333408831410494114245, 9.000644041338349206828923104341, 9.360930401498180975205138352635, 9.761758139217697478982108964917
