| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 3·7-s + 8-s − 2·9-s + 11-s − 12-s + 4·13-s + 3·14-s + 16-s + 3·17-s − 2·18-s − 5·19-s − 3·21-s + 22-s + 4·23-s − 24-s + 4·26-s + 5·27-s + 3·28-s + 5·29-s + 7·31-s + 32-s − 33-s + 3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.13·7-s + 0.353·8-s − 2/3·9-s + 0.301·11-s − 0.288·12-s + 1.10·13-s + 0.801·14-s + 1/4·16-s + 0.727·17-s − 0.471·18-s − 1.14·19-s − 0.654·21-s + 0.213·22-s + 0.834·23-s − 0.204·24-s + 0.784·26-s + 0.962·27-s + 0.566·28-s + 0.928·29-s + 1.25·31-s + 0.176·32-s − 0.174·33-s + 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.074912862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.074912862\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−10.87859972852879920732148596028, −10.41613231225318065197074702698, −8.717961421562585762296647152002, −8.242907688677664015983227610335, −6.89040237899544266304191025391, −6.04641270241765746429835400460, −5.18711436468138586194772102374, −4.31835852121277184532189905403, −3.01207625761226941022973660182, −1.40662093843722554519102852406,
1.40662093843722554519102852406, 3.01207625761226941022973660182, 4.31835852121277184532189905403, 5.18711436468138586194772102374, 6.04641270241765746429835400460, 6.89040237899544266304191025391, 8.242907688677664015983227610335, 8.717961421562585762296647152002, 10.41613231225318065197074702698, 10.87859972852879920732148596028
