| L(s) = 1 | + 4-s − 2·7-s − 3·9-s + 4·13-s + 16-s − 12·19-s + 6·25-s − 2·28-s − 4·31-s − 3·36-s + 20·37-s − 16·43-s + 3·49-s + 4·52-s − 28·61-s + 6·63-s + 64-s − 24·67-s + 8·73-s − 12·76-s + 9·81-s − 8·91-s − 28·97-s + 6·100-s + 36·103-s − 28·109-s − 2·112-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.755·7-s − 9-s + 1.10·13-s + 1/4·16-s − 2.75·19-s + 6/5·25-s − 0.377·28-s − 0.718·31-s − 1/2·36-s + 3.28·37-s − 2.43·43-s + 3/7·49-s + 0.554·52-s − 3.58·61-s + 0.755·63-s + 1/8·64-s − 2.93·67-s + 0.936·73-s − 1.37·76-s + 81-s − 0.838·91-s − 2.84·97-s + 3/5·100-s + 3.54·103-s − 2.68·109-s − 0.188·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−8.907037361397106316846384330744, −8.383894826086909083235343969228, −7.951642576699392563387512465158, −7.41620352139467519733544989343, −6.62445475159785453495873048483, −6.22872197038012242892904428945, −6.22501908736119603038057345164, −5.57547584921431243576094737475, −4.59118714632791292827191258849, −4.35107164128550817055960303512, −3.44639178218693714912433679777, −2.98342596760904293099474363551, −2.38911105843277843359160362418, −1.46845333940904453855522694316, 0,
1.46845333940904453855522694316, 2.38911105843277843359160362418, 2.98342596760904293099474363551, 3.44639178218693714912433679777, 4.35107164128550817055960303512, 4.59118714632791292827191258849, 5.57547584921431243576094737475, 6.22501908736119603038057345164, 6.22872197038012242892904428945, 6.62445475159785453495873048483, 7.41620352139467519733544989343, 7.951642576699392563387512465158, 8.383894826086909083235343969228, 8.907037361397106316846384330744
