| L(s) = 1 | + 3-s − 4-s + 5-s + 9-s − 12-s + 15-s − 2·17-s + 19-s − 20-s + 25-s + 2·27-s − 2·29-s − 36-s − 2·41-s + 45-s − 2·51-s + 53-s + 57-s − 59-s − 60-s + 64-s + 2·68-s + 4·71-s + 75-s − 76-s + 79-s + 2·81-s + ⋯ |
| L(s) = 1 | + 3-s − 4-s + 5-s + 9-s − 12-s + 15-s − 2·17-s + 19-s − 20-s + 25-s + 2·27-s − 2·29-s − 36-s − 2·41-s + 45-s − 2·51-s + 53-s + 57-s − 59-s − 60-s + 64-s + 2·68-s + 4·71-s + 75-s − 76-s + 79-s + 2·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8357881 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8357881 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.743775776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.743775776\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 59 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$ | \( ( 1 - T )^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.151552720672709293823699975324, −8.823573607109316093543880801994, −8.502879611332539410857679956041, −8.239776327952519283819585513749, −7.68672783402161022958912402917, −7.25128782032203409565381611584, −6.88335806846954413555967232607, −6.46750471128710390688533075515, −6.35369637138782202490704415144, −5.35434777230991222905767567092, −5.30286605319036745147130926708, −4.93845402367619572127930918241, −4.33709686595721967412685829130, −4.12937392474560267474951653692, −3.42596919566228130836220305286, −3.25189859017203059997583315054, −2.44134576684263061045762235445, −2.09395922604873230202677430274, −1.72631009811630911762167908708, −0.799467565213600756285561548217,
0.799467565213600756285561548217, 1.72631009811630911762167908708, 2.09395922604873230202677430274, 2.44134576684263061045762235445, 3.25189859017203059997583315054, 3.42596919566228130836220305286, 4.12937392474560267474951653692, 4.33709686595721967412685829130, 4.93845402367619572127930918241, 5.30286605319036745147130926708, 5.35434777230991222905767567092, 6.35369637138782202490704415144, 6.46750471128710390688533075515, 6.88335806846954413555967232607, 7.25128782032203409565381611584, 7.68672783402161022958912402917, 8.239776327952519283819585513749, 8.502879611332539410857679956041, 8.823573607109316093543880801994, 9.151552720672709293823699975324
