| L(s) = 1 | − 3-s + 2·7-s − 2·9-s − 12·13-s − 16·19-s − 2·21-s − 6·25-s + 5·27-s − 8·31-s − 2·37-s + 12·39-s + 4·43-s − 11·49-s + 16·57-s + 8·61-s − 4·63-s + 14·73-s + 6·75-s + 81-s − 24·91-s + 8·93-s − 16·97-s + 4·103-s + 32·109-s + 2·111-s + 24·117-s − 21·121-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s − 2/3·9-s − 3.32·13-s − 3.67·19-s − 0.436·21-s − 6/5·25-s + 0.962·27-s − 1.43·31-s − 0.328·37-s + 1.92·39-s + 0.609·43-s − 1.57·49-s + 2.11·57-s + 1.02·61-s − 0.503·63-s + 1.63·73-s + 0.692·75-s + 1/9·81-s − 2.51·91-s + 0.829·93-s − 1.62·97-s + 0.394·103-s + 3.06·109-s + 0.189·111-s + 2.21·117-s − 1.90·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 788544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 788544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 37 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
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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
−7.78790307165276969482845597819, −7.36881114003389544173968798730, −6.93610596286674052907325458245, −6.40622823243653800847643763088, −6.09037614955138614512431406118, −5.35667365720479356016286420004, −5.06148787959059824431190870012, −4.67375669110578511814790030096, −4.21402845745683233262500412192, −3.62376221698954049890975125760, −2.50673634667839885265978774188, −2.32752817979750411282892328426, −1.85774356352190049046929099208, 0, 0,
1.85774356352190049046929099208, 2.32752817979750411282892328426, 2.50673634667839885265978774188, 3.62376221698954049890975125760, 4.21402845745683233262500412192, 4.67375669110578511814790030096, 5.06148787959059824431190870012, 5.35667365720479356016286420004, 6.09037614955138614512431406118, 6.40622823243653800847643763088, 6.93610596286674052907325458245, 7.36881114003389544173968798730, 7.78790307165276969482845597819
