| L(s) = 1 | + 2·2-s + 3·4-s − 4·7-s + 4·8-s − 8·14-s + 5·16-s − 25-s − 12·28-s − 18·29-s + 6·32-s − 2·37-s + 16·43-s + 9·49-s − 2·50-s + 12·53-s − 16·56-s − 36·58-s + 7·64-s − 8·67-s + 24·71-s − 4·74-s − 32·79-s + 32·86-s + 18·98-s − 3·100-s + 24·106-s − 24·107-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 1.51·7-s + 1.41·8-s − 2.13·14-s + 5/4·16-s − 1/5·25-s − 2.26·28-s − 3.34·29-s + 1.06·32-s − 0.328·37-s + 2.43·43-s + 9/7·49-s − 0.282·50-s + 1.64·53-s − 2.13·56-s − 4.72·58-s + 7/8·64-s − 0.977·67-s + 2.84·71-s − 0.464·74-s − 3.60·79-s + 3.45·86-s + 1.81·98-s − 0.299·100-s + 2.33·106-s − 2.32·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.44339497869433550548235759101, −7.24345462277249174674167705434, −6.90845090857180264160912820283, −6.18735538198950166736525445007, −6.02284256820067515495072792009, −5.49090131952205180218546644094, −5.34577031750497924394207453637, −4.47042208842063244222746543662, −3.98303998918789627944532964156, −3.69589098584017718744945020548, −3.31148376843672612581522283320, −2.48651933877400261703339856802, −2.30016225881827778465495235815, −1.28744325263276944113521047912, 0,
1.28744325263276944113521047912, 2.30016225881827778465495235815, 2.48651933877400261703339856802, 3.31148376843672612581522283320, 3.69589098584017718744945020548, 3.98303998918789627944532964156, 4.47042208842063244222746543662, 5.34577031750497924394207453637, 5.49090131952205180218546644094, 6.02284256820067515495072792009, 6.18735538198950166736525445007, 6.90845090857180264160912820283, 7.24345462277249174674167705434, 7.44339497869433550548235759101
