| L(s) = 1 | + 3·2-s − 3-s + 4·4-s − 3·6-s + 3·7-s + 3·8-s + 6·11-s − 4·12-s + 2·13-s + 9·14-s + 3·16-s + 6·17-s + 3·19-s − 3·21-s + 18·22-s − 6·23-s − 3·24-s + 6·26-s + 27-s + 12·28-s + 6·29-s + 6·32-s − 6·33-s + 18·34-s + 12·37-s + 9·38-s − 2·39-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 0.577·3-s + 2·4-s − 1.22·6-s + 1.13·7-s + 1.06·8-s + 1.80·11-s − 1.15·12-s + 0.554·13-s + 2.40·14-s + 3/4·16-s + 1.45·17-s + 0.688·19-s − 0.654·21-s + 3.83·22-s − 1.25·23-s − 0.612·24-s + 1.17·26-s + 0.192·27-s + 2.26·28-s + 1.11·29-s + 1.06·32-s − 1.04·33-s + 3.08·34-s + 1.97·37-s + 1.45·38-s − 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.843617007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.843617007\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 6 T + 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 18 T + 167 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 71 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 6 T + 101 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 12 T + 145 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−10.14680117524389124301079234360, −9.988232094334537320592056594805, −9.562651510540609225240983901916, −8.932984431179282466936062387372, −8.340408706124505785141578713123, −8.133009226825014810609336410098, −7.52094940499952022564580548254, −7.18905125717837775983348241682, −6.27866975632635721103642602270, −6.24473490827970483894395776832, −5.86669432097327649595771951187, −5.37432467929076472907416645543, −4.79378626973803363843434364099, −4.59820276814727245348684063064, −4.12596854702059674981854834713, −3.59680014972485563724286043485, −3.29436043286518922483649953676, −2.50809575543344683957342221366, −1.34983603903568384525924615360, −1.27350253094888388939779283647,
1.27350253094888388939779283647, 1.34983603903568384525924615360, 2.50809575543344683957342221366, 3.29436043286518922483649953676, 3.59680014972485563724286043485, 4.12596854702059674981854834713, 4.59820276814727245348684063064, 4.79378626973803363843434364099, 5.37432467929076472907416645543, 5.86669432097327649595771951187, 6.24473490827970483894395776832, 6.27866975632635721103642602270, 7.18905125717837775983348241682, 7.52094940499952022564580548254, 8.133009226825014810609336410098, 8.340408706124505785141578713123, 8.932984431179282466936062387372, 9.562651510540609225240983901916, 9.988232094334537320592056594805, 10.14680117524389124301079234360
