| L(s) = 1 | − 2-s + 4-s − 4·5-s + 4·7-s − 8-s − 3·9-s + 4·10-s − 2·11-s − 2·13-s − 4·14-s + 16-s + 2·17-s + 3·18-s + 2·19-s − 4·20-s + 2·22-s + 11·25-s + 2·26-s + 4·28-s + 2·29-s − 32-s − 2·34-s − 16·35-s − 3·36-s + 4·37-s − 2·38-s + 4·40-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.78·5-s + 1.51·7-s − 0.353·8-s − 9-s + 1.26·10-s − 0.603·11-s − 0.554·13-s − 1.06·14-s + 1/4·16-s + 0.485·17-s + 0.707·18-s + 0.458·19-s − 0.894·20-s + 0.426·22-s + 11/5·25-s + 0.392·26-s + 0.755·28-s + 0.371·29-s − 0.176·32-s − 0.342·34-s − 2.70·35-s − 1/2·36-s + 0.657·37-s − 0.324·38-s + 0.632·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1058 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1058 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7583301280\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7583301280\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−9.947689550336840395250376815301, −8.683650556938610005541061706290, −8.159724093271066632099122061158, −7.76953499301176804540241574451, −6.97395956350401904764981202712, −5.45714573817728122372138066179, −4.72575079143831508702325515450, −3.56810168580414870732367804186, −2.44935456312664529103463287959, −0.73237197917342842336436187115,
0.73237197917342842336436187115, 2.44935456312664529103463287959, 3.56810168580414870732367804186, 4.72575079143831508702325515450, 5.45714573817728122372138066179, 6.97395956350401904764981202712, 7.76953499301176804540241574451, 8.159724093271066632099122061158, 8.683650556938610005541061706290, 9.947689550336840395250376815301
