L(s) = 1 | + 3·3-s − 4-s + 5-s + 5·7-s + 6·9-s − 3·12-s + 3·15-s − 3·16-s − 20-s + 15·21-s − 7·25-s + 9·27-s − 5·28-s + 3·29-s − 31-s + 5·35-s − 6·36-s − 10·43-s + 6·45-s − 9·48-s + 7·49-s − 3·60-s + 7·61-s + 30·63-s + 7·64-s − 7·71-s − 2·73-s + ⋯ |
L(s) = 1 | + 1.73·3-s − 1/2·4-s + 0.447·5-s + 1.88·7-s + 2·9-s − 0.866·12-s + 0.774·15-s − 3/4·16-s − 0.223·20-s + 3.27·21-s − 7/5·25-s + 1.73·27-s − 0.944·28-s + 0.557·29-s − 0.179·31-s + 0.845·35-s − 36-s − 1.52·43-s + 0.894·45-s − 1.29·48-s + 49-s − 0.387·60-s + 0.896·61-s + 3.77·63-s + 7/8·64-s − 0.830·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106929 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106929 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.279880432\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.279880432\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 109 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 21 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 39 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 67 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 61 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
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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
−9.560697429379555000562187686781, −8.767733863324597346413621405042, −8.486933386545845736318030881707, −8.336855354557852660287194185554, −7.68000845889351253569006811313, −7.31007270777464263030885865926, −6.71003183621670025346877681019, −5.86536693412791737873523310255, −5.20200887007111970766390208538, −4.61434543184931961903324985364, −4.26826761650802663793043459728, −3.57128018467403453546884343802, −2.73640559214550749996189750007, −1.96847945163320857683932955343, −1.57028550461269840502407188614,
1.57028550461269840502407188614, 1.96847945163320857683932955343, 2.73640559214550749996189750007, 3.57128018467403453546884343802, 4.26826761650802663793043459728, 4.61434543184931961903324985364, 5.20200887007111970766390208538, 5.86536693412791737873523310255, 6.71003183621670025346877681019, 7.31007270777464263030885865926, 7.68000845889351253569006811313, 8.336855354557852660287194185554, 8.486933386545845736318030881707, 8.767733863324597346413621405042, 9.560697429379555000562187686781