L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s − 2·9-s − 10-s + 12-s + 5·13-s + 14-s − 15-s + 16-s + 6·17-s − 2·18-s + 5·19-s − 20-s + 21-s − 3·23-s + 24-s + 25-s + 5·26-s − 5·27-s + 28-s − 30-s − 10·31-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.316·10-s + 0.288·12-s + 1.38·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.471·18-s + 1.14·19-s − 0.223·20-s + 0.218·21-s − 0.625·23-s + 0.204·24-s + 1/5·25-s + 0.980·26-s − 0.962·27-s + 0.188·28-s − 0.182·30-s − 1.79·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.328783787\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.328783787\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−7.77885660914471464515671962142, −7.27102357623235489491631904304, −6.21388324451130204404891198367, −5.61167526510904250529201560390, −5.14635085729982464772585355260, −3.90695180116991245126928471775, −3.61741183568389224115075321549, −2.92416460353575086244737076715, −1.90142706031601762055437172017, −0.923730773406296477096509948120,
0.923730773406296477096509948120, 1.90142706031601762055437172017, 2.92416460353575086244737076715, 3.61741183568389224115075321549, 3.90695180116991245126928471775, 5.14635085729982464772585355260, 5.61167526510904250529201560390, 6.21388324451130204404891198367, 7.27102357623235489491631904304, 7.77885660914471464515671962142