| L(s) = 1 | + 3-s − 2·5-s + 2·7-s + 9-s + 5·11-s + 3·13-s − 2·15-s − 3·17-s − 2·19-s + 2·21-s + 23-s − 25-s + 27-s + 5·31-s + 5·33-s − 4·35-s + 8·37-s + 3·39-s − 7·41-s + 43-s − 2·45-s + 8·47-s − 3·49-s − 3·51-s + 3·53-s − 10·55-s − 2·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 0.755·7-s + 1/3·9-s + 1.50·11-s + 0.832·13-s − 0.516·15-s − 0.727·17-s − 0.458·19-s + 0.436·21-s + 0.208·23-s − 1/5·25-s + 0.192·27-s + 0.898·31-s + 0.870·33-s − 0.676·35-s + 1.31·37-s + 0.480·39-s − 1.09·41-s + 0.152·43-s − 0.298·45-s + 1.16·47-s − 3/7·49-s − 0.420·51-s + 0.412·53-s − 1.34·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.276195795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.276195795\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−8.978009802352649671808602680382, −8.311202822356102145576413186265, −7.80436893404074773763481968895, −6.78461901069866544425570526686, −6.19603689777594065734989014157, −4.79535808046119236916099761001, −4.10319001261157529164243453614, −3.50887355939377684370334646650, −2.16208352631894123622226457901, −1.04596298856260690087985554195,
1.04596298856260690087985554195, 2.16208352631894123622226457901, 3.50887355939377684370334646650, 4.10319001261157529164243453614, 4.79535808046119236916099761001, 6.19603689777594065734989014157, 6.78461901069866544425570526686, 7.80436893404074773763481968895, 8.311202822356102145576413186265, 8.978009802352649671808602680382
