| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s − 11-s + 12-s + 13-s + 14-s + 16-s − 17-s − 18-s − 2·19-s − 21-s + 22-s + 3·23-s − 24-s − 26-s + 27-s − 28-s + 5·29-s + 31-s − 32-s − 33-s + 34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.458·19-s − 0.218·21-s + 0.213·22-s + 0.625·23-s − 0.204·24-s − 0.196·26-s + 0.192·27-s − 0.188·28-s + 0.928·29-s + 0.179·31-s − 0.176·32-s − 0.174·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.497043678\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.497043678\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.58787873135263, −12.02449366638238, −11.50399605598141, −11.04062579298377, −10.69086654079748, −10.14607351707171, −9.694465691492973, −9.369472057313554, −8.908822152559213, −8.338270579581170, −8.032978368810963, −7.686943671436069, −6.909689147171097, −6.659581525375520, −6.173970023908651, −5.663371222132312, −4.822398457827290, −4.588616921725590, −3.831482915879908, −3.284160412810409, −2.761014183220083, −2.423552697583149, −1.615303143787729, −1.150700685759642, −0.3546449168431864,
0.3546449168431864, 1.150700685759642, 1.615303143787729, 2.423552697583149, 2.761014183220083, 3.284160412810409, 3.831482915879908, 4.588616921725590, 4.822398457827290, 5.663371222132312, 6.173970023908651, 6.659581525375520, 6.909689147171097, 7.686943671436069, 8.032978368810963, 8.338270579581170, 8.908822152559213, 9.369472057313554, 9.694465691492973, 10.14607351707171, 10.69086654079748, 11.04062579298377, 11.50399605598141, 12.02449366638238, 12.58787873135263
