| L(s) = 1 | − 4·7-s − 9-s − 4·17-s − 4·23-s − 25-s + 8·31-s + 20·41-s + 4·47-s − 2·49-s + 4·63-s + 16·71-s + 12·73-s + 32·79-s + 81-s + 20·89-s − 12·97-s − 12·103-s − 4·113-s + 16·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·153-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1/3·9-s − 0.970·17-s − 0.834·23-s − 1/5·25-s + 1.43·31-s + 3.12·41-s + 0.583·47-s − 2/7·49-s + 0.503·63-s + 1.89·71-s + 1.40·73-s + 3.60·79-s + 1/9·81-s + 2.11·89-s − 1.21·97-s − 1.18·103-s − 0.376·113-s + 1.46·119-s + 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.323·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.913138181\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.913138181\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−8.774952585729965456129835243451, −8.033509809642357855705635274075, −7.993912482476684355209898096027, −7.83256393604755001192482682697, −6.99942461829639485527240670252, −6.76742743840112987074920504934, −6.36100173357021849664235289123, −6.32539431110793658148666083920, −5.62232690846816836210592116059, −5.58083512635256841301766302033, −4.73637392265189121320659849815, −4.55047325233072525417247384495, −4.00343606549441224667644928499, −3.65577386419758656116920396697, −3.19988952236857052127572822872, −2.74643693783960879489484546063, −2.24930454546685715341295803470, −1.99369228606659514884963840121, −0.78859289201617975783698704764, −0.55929500020439937515470740784,
0.55929500020439937515470740784, 0.78859289201617975783698704764, 1.99369228606659514884963840121, 2.24930454546685715341295803470, 2.74643693783960879489484546063, 3.19988952236857052127572822872, 3.65577386419758656116920396697, 4.00343606549441224667644928499, 4.55047325233072525417247384495, 4.73637392265189121320659849815, 5.58083512635256841301766302033, 5.62232690846816836210592116059, 6.32539431110793658148666083920, 6.36100173357021849664235289123, 6.76742743840112987074920504934, 6.99942461829639485527240670252, 7.83256393604755001192482682697, 7.993912482476684355209898096027, 8.033509809642357855705635274075, 8.774952585729965456129835243451
