| L(s) = 1 | − 2-s − 3-s + 4-s − 3·5-s + 6-s − 8-s + 9-s + 3·10-s − 12-s + 3·15-s + 16-s − 18-s − 3·20-s − 4·23-s + 24-s + 2·25-s − 27-s + 3·29-s − 3·30-s − 32-s + 36-s + 3·40-s − 4·43-s − 3·45-s + 4·46-s − 8·47-s − 48-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.948·10-s − 0.288·12-s + 0.774·15-s + 1/4·16-s − 0.235·18-s − 0.670·20-s − 0.834·23-s + 0.204·24-s + 2/5·25-s − 0.192·27-s + 0.557·29-s − 0.547·30-s − 0.176·32-s + 1/6·36-s + 0.474·40-s − 0.609·43-s − 0.447·45-s + 0.589·46-s − 1.16·47-s − 0.144·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| good | 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.328271771673691702127063606955, −7.917007245101163531970345056430, −7.35070963643901245623154602985, −7.11239089095483680348031787161, −6.51918570496581688607568201726, −6.13681234160980735696967522151, −5.49849711956353977302820263044, −5.01846427260912693962721867921, −4.38149088188910897304792349877, −3.88434519751423725949927238797, −3.46919894893418346345427789874, −2.65443210197633591425468016106, −1.91012298358415041488379971630, −0.919393510240047424904618514228, 0,
0.919393510240047424904618514228, 1.91012298358415041488379971630, 2.65443210197633591425468016106, 3.46919894893418346345427789874, 3.88434519751423725949927238797, 4.38149088188910897304792349877, 5.01846427260912693962721867921, 5.49849711956353977302820263044, 6.13681234160980735696967522151, 6.51918570496581688607568201726, 7.11239089095483680348031787161, 7.35070963643901245623154602985, 7.917007245101163531970345056430, 8.328271771673691702127063606955
