| 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
| \(L(1)\) |
\(\approx\) |
\(2.575199058\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.575199058\) |
| \(L(\frac{3}{2})\) |
|
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
−8.642576883784357485317719719519, −8.387235697683322779103248953900, −7.77218661530784050384824754408, −7.71194726067683544048807456397, −7.28019205864201834164239587742, −6.96109599096774671422589540585, −6.40975104147214998069390424067, −6.04827290126061187767847684858, −5.60384106266487157639730916219, −5.38824731940162947893244568320, −4.75690357223641027068115050912, −4.59155090754691174682965076314, −4.18637422964580104704780865338, −3.78332668661248093324804996735, −2.94163458531139799961089589238, −2.91736997855372527311901628097, −1.99953921150064950113224090286, −1.89076620547004937057939856396, −1.21078594327797846718405580473, −0.48473409333229581747745503111,
0.48473409333229581747745503111, 1.21078594327797846718405580473, 1.89076620547004937057939856396, 1.99953921150064950113224090286, 2.91736997855372527311901628097, 2.94163458531139799961089589238, 3.78332668661248093324804996735, 4.18637422964580104704780865338, 4.59155090754691174682965076314, 4.75690357223641027068115050912, 5.38824731940162947893244568320, 5.60384106266487157639730916219, 6.04827290126061187767847684858, 6.40975104147214998069390424067, 6.96109599096774671422589540585, 7.28019205864201834164239587742, 7.71194726067683544048807456397, 7.77218661530784050384824754408, 8.387235697683322779103248953900, 8.642576883784357485317719719519
