| L(s) = 1 | − 3·3-s − 6·5-s − 6·7-s + 6·9-s − 3·11-s − 4·13-s + 18·15-s + 18·21-s + 19·25-s − 9·27-s − 6·29-s + 9·33-s + 36·35-s + 4·37-s + 12·39-s − 9·41-s + 9·43-s − 36·45-s − 12·47-s + 17·49-s + 18·55-s − 15·59-s − 8·61-s − 36·63-s + 24·65-s − 15·67-s − 12·71-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 2.68·5-s − 2.26·7-s + 2·9-s − 0.904·11-s − 1.10·13-s + 4.64·15-s + 3.92·21-s + 19/5·25-s − 1.73·27-s − 1.11·29-s + 1.56·33-s + 6.08·35-s + 0.657·37-s + 1.92·39-s − 1.40·41-s + 1.37·43-s − 5.36·45-s − 1.75·47-s + 17/7·49-s + 2.42·55-s − 1.95·59-s − 1.02·61-s − 4.53·63-s + 2.97·65-s − 1.83·67-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 41 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 9 T + 70 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 15 T + 166 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 6 T + 91 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 13 T + 72 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
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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
−12.53738723894124325322831459464, −12.30401005689130514723881916050, −11.81104424158415606724684820311, −11.61037786181095395031438337570, −10.76713271626389253569414382272, −10.62173271796120513717245260138, −9.882101617114231288055950086789, −9.455304537187462249333404721519, −8.631598278225329330423059223842, −7.71317973727378454066988344492, −7.34671063527856289449570099151, −7.14890516216849682363567770269, −6.21303297723190117732479373272, −5.89872616341568826783990414528, −4.67338578996700525503174029078, −4.57699756012352974507332297910, −3.48837582993728922263359163609, −3.10905564533405997568619502780, 0, 0,
3.10905564533405997568619502780, 3.48837582993728922263359163609, 4.57699756012352974507332297910, 4.67338578996700525503174029078, 5.89872616341568826783990414528, 6.21303297723190117732479373272, 7.14890516216849682363567770269, 7.34671063527856289449570099151, 7.71317973727378454066988344492, 8.631598278225329330423059223842, 9.455304537187462249333404721519, 9.882101617114231288055950086789, 10.62173271796120513717245260138, 10.76713271626389253569414382272, 11.61037786181095395031438337570, 11.81104424158415606724684820311, 12.30401005689130514723881916050, 12.53738723894124325322831459464
