| L(s) = 1 | − 3·2-s + 3·4-s − 3·7-s + 3·8-s − 9-s − 11-s + 9·14-s − 13·16-s + 3·18-s + 3·22-s − 2·23-s + 25-s − 9·28-s − 12·29-s + 15·32-s − 3·36-s − 10·37-s − 12·43-s − 3·44-s + 6·46-s + 2·49-s − 3·50-s − 6·53-s − 9·56-s + 36·58-s + 3·63-s + 3·64-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3/2·4-s − 1.13·7-s + 1.06·8-s − 1/3·9-s − 0.301·11-s + 2.40·14-s − 3.25·16-s + 0.707·18-s + 0.639·22-s − 0.417·23-s + 1/5·25-s − 1.70·28-s − 2.22·29-s + 2.65·32-s − 1/2·36-s − 1.64·37-s − 1.82·43-s − 0.452·44-s + 0.884·46-s + 2/7·49-s − 0.424·50-s − 0.824·53-s − 1.20·56-s + 4.72·58-s + 0.377·63-s + 3/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 586971 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 586971 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 129 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 41 T^{2} + p^{2} T^{4} \) |
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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
−8.033985119680657758809722954176, −7.74369015491182541402709500332, −7.30968188461232586213102636035, −6.83272202833821117799248538393, −6.42109925053112637840628694456, −5.77853288285851813126274785821, −5.18463249224441976577017374083, −4.79570112335091341268005650173, −3.90161152981856838601851497832, −3.57535011227738440343483981990, −2.81177726958092500766016878480, −1.92461272762362572325073508941, −1.40406174238153522750275890113, 0, 0,
1.40406174238153522750275890113, 1.92461272762362572325073508941, 2.81177726958092500766016878480, 3.57535011227738440343483981990, 3.90161152981856838601851497832, 4.79570112335091341268005650173, 5.18463249224441976577017374083, 5.77853288285851813126274785821, 6.42109925053112637840628694456, 6.83272202833821117799248538393, 7.30968188461232586213102636035, 7.74369015491182541402709500332, 8.033985119680657758809722954176
