| L(s) = 1 | − 2-s + 3-s + 4-s + 3·5-s − 6-s − 7-s − 8-s + 9-s − 3·10-s + 12-s + 13-s + 14-s + 3·15-s + 16-s + 17-s − 18-s + 6·19-s + 3·20-s − 21-s + 4·23-s − 24-s + 4·25-s − 26-s + 27-s − 28-s − 5·29-s − 3·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.288·12-s + 0.277·13-s + 0.267·14-s + 0.774·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 1.37·19-s + 0.670·20-s − 0.218·21-s + 0.834·23-s − 0.204·24-s + 4/5·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s − 0.928·29-s − 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9282 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9282 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.722321467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.722321467\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−7.76526262572370164351628103956, −7.05601244261590775053658916568, −6.52793463919209635220403328645, −5.62878324644083624811438742878, −5.29323664414887359275222958151, −4.03730666191845187401185334751, −3.13546017853523036978628034530, −2.54862016430903188279373021562, −1.67496376787686394768634268150, −0.905485897273856916174751314634,
0.905485897273856916174751314634, 1.67496376787686394768634268150, 2.54862016430903188279373021562, 3.13546017853523036978628034530, 4.03730666191845187401185334751, 5.29323664414887359275222958151, 5.62878324644083624811438742878, 6.52793463919209635220403328645, 7.05601244261590775053658916568, 7.76526262572370164351628103956
