| L(s) = 1 | − 3-s + 9-s − 2·13-s − 4·16-s − 2·19-s − 6·25-s − 27-s − 18·31-s + 6·37-s + 2·39-s + 10·43-s + 4·48-s + 2·57-s − 20·61-s − 10·67-s + 6·73-s + 6·75-s − 2·79-s + 81-s + 18·93-s + 12·97-s + 14·103-s + 18·109-s − 6·111-s − 2·117-s − 18·121-s + 127-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.554·13-s − 16-s − 0.458·19-s − 6/5·25-s − 0.192·27-s − 3.23·31-s + 0.986·37-s + 0.320·39-s + 1.52·43-s + 0.577·48-s + 0.264·57-s − 2.56·61-s − 1.22·67-s + 0.702·73-s + 0.692·75-s − 0.225·79-s + 1/9·81-s + 1.86·93-s + 1.21·97-s + 1.37·103-s + 1.72·109-s − 0.569·111-s − 0.184·117-s − 1.63·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64827 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64827 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−9.532401741076686225901695719699, −9.092563983560312555964020990816, −8.970613305827843910423804944183, −7.920026489093004352764513852408, −7.42251681884249141719743547364, −7.29485403540511127428179965078, −6.33310252534575837626677109971, −6.01621490934601452163404663718, −5.40855511243246313730427714163, −4.73946994522750223509171420340, −4.18739539622885934722496613081, −3.54901712994340686137812719330, −2.47062022887811000287520417928, −1.75196450155362895800400842510, 0,
1.75196450155362895800400842510, 2.47062022887811000287520417928, 3.54901712994340686137812719330, 4.18739539622885934722496613081, 4.73946994522750223509171420340, 5.40855511243246313730427714163, 6.01621490934601452163404663718, 6.33310252534575837626677109971, 7.29485403540511127428179965078, 7.42251681884249141719743547364, 7.920026489093004352764513852408, 8.970613305827843910423804944183, 9.092563983560312555964020990816, 9.532401741076686225901695719699
