| L(s) = 1 | − 2-s + 3-s − 4-s − 6-s + 3·8-s + 9-s + 8·11-s − 12-s − 4·13-s − 16-s − 18-s − 8·22-s + 3·24-s − 6·25-s + 4·26-s + 27-s − 5·32-s + 8·33-s − 36-s + 12·37-s − 4·39-s − 8·44-s − 48-s + 49-s + 6·50-s + 4·52-s − 54-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 1.06·8-s + 1/3·9-s + 2.41·11-s − 0.288·12-s − 1.10·13-s − 1/4·16-s − 0.235·18-s − 1.70·22-s + 0.612·24-s − 6/5·25-s + 0.784·26-s + 0.192·27-s − 0.883·32-s + 1.39·33-s − 1/6·36-s + 1.97·37-s − 0.640·39-s − 1.20·44-s − 0.144·48-s + 1/7·49-s + 0.848·50-s + 0.554·52-s − 0.136·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21168 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21168 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9399498356\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9399498356\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−10.64068584777438676509120304615, −9.857567892373655758925106479846, −9.724661210726173973340366981146, −9.306486168015385400395632025578, −8.672365196683144779961098572501, −8.351153010775658504468507691128, −7.45452630662509700459332399892, −7.25047783802838427330028450541, −6.43539728173776997986575871832, −5.76945209288606009941542010236, −4.75834670654208540745088589071, −4.13559084050773741974089362056, −3.71279869000131289976682575519, −2.38203049551357769485459868076, −1.27003349896142462152543818564,
1.27003349896142462152543818564, 2.38203049551357769485459868076, 3.71279869000131289976682575519, 4.13559084050773741974089362056, 4.75834670654208540745088589071, 5.76945209288606009941542010236, 6.43539728173776997986575871832, 7.25047783802838427330028450541, 7.45452630662509700459332399892, 8.351153010775658504468507691128, 8.672365196683144779961098572501, 9.306486168015385400395632025578, 9.724661210726173973340366981146, 9.857567892373655758925106479846, 10.64068584777438676509120304615
