| L(s) = 1 | − 2-s − 4-s − 7-s + 8-s + 4·9-s + 14-s + 3·16-s − 4·18-s + 3·23-s + 8·25-s + 28-s + 3·29-s − 3·32-s − 4·36-s + 4·37-s − 11·43-s − 3·46-s − 6·49-s − 8·50-s + 15·53-s − 56-s − 3·58-s − 4·63-s − 5·64-s + 67-s − 9·71-s + 4·72-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.377·7-s + 0.353·8-s + 4/3·9-s + 0.267·14-s + 3/4·16-s − 0.942·18-s + 0.625·23-s + 8/5·25-s + 0.188·28-s + 0.557·29-s − 0.530·32-s − 2/3·36-s + 0.657·37-s − 1.67·43-s − 0.442·46-s − 6/7·49-s − 1.13·50-s + 2.06·53-s − 0.133·56-s − 0.393·58-s − 0.503·63-s − 5/8·64-s + 0.122·67-s − 1.06·71-s + 0.471·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10682 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10682 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6395679587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6395679587\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 109 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 101 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
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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
−11.45015467590693421164346509702, −10.59531048952671103438107516496, −10.14148718376678192266383006025, −9.937215580517584118283960355209, −9.099653024909612921880514011235, −8.815697466748258356959404572848, −8.156693826142832276194917034905, −7.42648111208650801711524598128, −6.89148387018769867154776861331, −6.29813419050289528037616837827, −5.25365473110579765631698501017, −4.66837443786080150871832452155, −3.85055513300718885076242085541, −2.89009034690832333624089485932, −1.24030723994401829440977114502,
1.24030723994401829440977114502, 2.89009034690832333624089485932, 3.85055513300718885076242085541, 4.66837443786080150871832452155, 5.25365473110579765631698501017, 6.29813419050289528037616837827, 6.89148387018769867154776861331, 7.42648111208650801711524598128, 8.156693826142832276194917034905, 8.815697466748258356959404572848, 9.099653024909612921880514011235, 9.937215580517584118283960355209, 10.14148718376678192266383006025, 10.59531048952671103438107516496, 11.45015467590693421164346509702
