| L(s) = 1 | − 3-s + 7-s + 9-s − 2·11-s + 13-s − 17-s − 4·19-s − 21-s − 7·23-s − 27-s + 29-s − 3·31-s + 2·33-s − 6·37-s − 39-s − 3·41-s + 43-s + 12·47-s + 49-s + 51-s + 11·53-s + 4·57-s + 3·59-s + 5·61-s + 63-s + 12·67-s + 7·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.603·11-s + 0.277·13-s − 0.242·17-s − 0.917·19-s − 0.218·21-s − 1.45·23-s − 0.192·27-s + 0.185·29-s − 0.538·31-s + 0.348·33-s − 0.986·37-s − 0.160·39-s − 0.468·41-s + 0.152·43-s + 1.75·47-s + 1/7·49-s + 0.140·51-s + 1.51·53-s + 0.529·57-s + 0.390·59-s + 0.640·61-s + 0.125·63-s + 1.46·67-s + 0.842·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.210251011\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.210251011\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−7.71933068321402287690494157096, −7.11649809968246956995011535930, −6.34012658169361350201138493842, −5.69878812775070818997246001336, −5.12054874122816893040014865857, −4.24649582829538177840313621233, −3.71583083361320935135400409279, −2.46051517989730097036730923408, −1.80901921449176103097191917242, −0.54388693290682410885311378219,
0.54388693290682410885311378219, 1.80901921449176103097191917242, 2.46051517989730097036730923408, 3.71583083361320935135400409279, 4.24649582829538177840313621233, 5.12054874122816893040014865857, 5.69878812775070818997246001336, 6.34012658169361350201138493842, 7.11649809968246956995011535930, 7.71933068321402287690494157096
