L(s) = 1 | + 3-s − 6·7-s + 9-s + 2·13-s − 4·16-s + 4·19-s − 6·21-s + 25-s + 27-s − 4·31-s − 6·37-s + 2·39-s − 8·43-s − 4·48-s + 13·49-s + 4·57-s − 22·61-s − 6·63-s − 8·67-s + 12·73-s + 75-s − 22·79-s + 81-s − 12·91-s − 4·93-s − 18·97-s + 8·103-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 2.26·7-s + 1/3·9-s + 0.554·13-s − 16-s + 0.917·19-s − 1.30·21-s + 1/5·25-s + 0.192·27-s − 0.718·31-s − 0.986·37-s + 0.320·39-s − 1.21·43-s − 0.577·48-s + 13/7·49-s + 0.529·57-s − 2.81·61-s − 0.755·63-s − 0.977·67-s + 1.40·73-s + 0.115·75-s − 2.47·79-s + 1/9·81-s − 1.25·91-s − 0.414·93-s − 1.82·97-s + 0.788·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114075 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114075 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
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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
−9.175134422877607002063538220098, −8.923978077954925138616609003816, −8.434475207546954053625988897044, −7.66337536315553181125482042712, −7.07755893352212952629549495812, −6.88984608666809355423599409217, −6.15614460284795576397102652158, −5.93954304028653148352357105381, −5.02803177361324098534837817934, −4.40005354130592369090367498749, −3.48200443418719418963780366073, −3.33489564696074388776632712954, −2.70048152954381141744823279421, −1.62636499494931476921849119993, 0,
1.62636499494931476921849119993, 2.70048152954381141744823279421, 3.33489564696074388776632712954, 3.48200443418719418963780366073, 4.40005354130592369090367498749, 5.02803177361324098534837817934, 5.93954304028653148352357105381, 6.15614460284795576397102652158, 6.88984608666809355423599409217, 7.07755893352212952629549495812, 7.66337536315553181125482042712, 8.434475207546954053625988897044, 8.923978077954925138616609003816, 9.175134422877607002063538220098