| L(s) = 1 | + 3-s + 5-s − 2·9-s + 3·11-s − 5·13-s + 15-s − 3·17-s + 2·19-s + 6·23-s + 25-s − 5·27-s + 3·29-s − 4·31-s + 3·33-s + 2·37-s − 5·39-s + 12·41-s + 10·43-s − 2·45-s + 9·47-s − 3·51-s + 12·53-s + 3·55-s + 2·57-s − 8·61-s − 5·65-s + 4·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 2/3·9-s + 0.904·11-s − 1.38·13-s + 0.258·15-s − 0.727·17-s + 0.458·19-s + 1.25·23-s + 1/5·25-s − 0.962·27-s + 0.557·29-s − 0.718·31-s + 0.522·33-s + 0.328·37-s − 0.800·39-s + 1.87·41-s + 1.52·43-s − 0.298·45-s + 1.31·47-s − 0.420·51-s + 1.64·53-s + 0.404·55-s + 0.264·57-s − 1.02·61-s − 0.620·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.391079284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.391079284\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−8.703677986065848624605724183186, −7.59679103544400835112633947355, −7.18840908028866321022100981317, −6.23576189288857341841080164821, −5.51580216499301883688612135353, −4.66901616031576415432379359350, −3.81894678634232741164227871759, −2.72282955438852544862637314814, −2.28641578958231941287903879491, −0.863841609060871512635709852891,
0.863841609060871512635709852891, 2.28641578958231941287903879491, 2.72282955438852544862637314814, 3.81894678634232741164227871759, 4.66901616031576415432379359350, 5.51580216499301883688612135353, 6.23576189288857341841080164821, 7.18840908028866321022100981317, 7.59679103544400835112633947355, 8.703677986065848624605724183186
