| L(s) = 1 | − 3-s − 3·4-s − 4·7-s + 9-s + 3·12-s + 5·16-s − 16·19-s + 4·21-s + 25-s − 27-s + 12·28-s + 2·31-s − 3·36-s + 16·37-s − 5·48-s − 2·49-s + 16·57-s − 28·61-s − 4·63-s − 3·64-s + 4·67-s − 32·73-s − 75-s + 48·76-s + 81-s − 12·84-s − 2·93-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 3/2·4-s − 1.51·7-s + 1/3·9-s + 0.866·12-s + 5/4·16-s − 3.67·19-s + 0.872·21-s + 1/5·25-s − 0.192·27-s + 2.26·28-s + 0.359·31-s − 1/2·36-s + 2.63·37-s − 0.721·48-s − 2/7·49-s + 2.11·57-s − 3.58·61-s − 0.503·63-s − 3/8·64-s + 0.488·67-s − 3.74·73-s − 0.115·75-s + 5.50·76-s + 1/9·81-s − 1.30·84-s − 0.207·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648675 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648675 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 31 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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
−7.84532589169742215840335224293, −7.68128969255594037111738820978, −6.77225946146642489618389989901, −6.36573875529349558957576705485, −6.13319511217733986766315838164, −5.87102863181679820049643029680, −4.85221518213525471918852830380, −4.62476804594428139253779251777, −4.15549500339935073670126654606, −3.85399261511576687175772764034, −2.96653770006789309828242375320, −2.48347553325427240017401962963, −1.38137651049014592943287717063, 0, 0,
1.38137651049014592943287717063, 2.48347553325427240017401962963, 2.96653770006789309828242375320, 3.85399261511576687175772764034, 4.15549500339935073670126654606, 4.62476804594428139253779251777, 4.85221518213525471918852830380, 5.87102863181679820049643029680, 6.13319511217733986766315838164, 6.36573875529349558957576705485, 6.77225946146642489618389989901, 7.68128969255594037111738820978, 7.84532589169742215840335224293
