| L(s) = 1 | + 2-s − 3-s + 4-s + 2·5-s − 6-s − 2·7-s + 8-s + 9-s + 2·10-s + 11-s − 12-s + 4·13-s − 2·14-s − 2·15-s + 16-s + 17-s + 18-s − 2·19-s + 2·20-s + 2·21-s + 22-s + 2·23-s − 24-s − 25-s + 4·26-s − 27-s − 2·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s − 0.288·12-s + 1.10·13-s − 0.534·14-s − 0.516·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.458·19-s + 0.447·20-s + 0.436·21-s + 0.213·22-s + 0.417·23-s − 0.204·24-s − 1/5·25-s + 0.784·26-s − 0.192·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.456721278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.456721278\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−19.96951418983534, −19.29090400124457, −18.53148258980714, −17.77692966869719, −17.08329698753501, −16.48604075540608, −15.75075562542054, −15.16575103537272, −14.08277312618761, −13.56072578142840, −13.01657433399401, −12.19829079961104, −11.53034116856518, −10.56094587925171, −10.06578207306856, −9.150940393068329, −8.169804735948645, −6.849408517496767, −6.294576530190562, −5.756033086840227, −4.708745458885183, −3.702091175650628, −2.600600053110328, −1.215135894462684,
1.215135894462684, 2.600600053110328, 3.702091175650628, 4.708745458885183, 5.756033086840227, 6.294576530190562, 6.849408517496767, 8.169804735948645, 9.150940393068329, 10.06578207306856, 10.56094587925171, 11.53034116856518, 12.19829079961104, 13.01657433399401, 13.56072578142840, 14.08277312618761, 15.16575103537272, 15.75075562542054, 16.48604075540608, 17.08329698753501, 17.77692966869719, 18.53148258980714, 19.29090400124457, 19.96951418983534
