| L(s) = 1 | − 5-s − 6·11-s + 4·13-s + 6·17-s − 2·19-s + 25-s − 6·29-s + 10·31-s + 2·37-s − 6·41-s − 4·43-s + 12·53-s + 6·55-s − 14·61-s − 4·65-s − 4·67-s − 6·71-s + 4·73-s − 16·79-s − 12·83-s − 6·85-s + 6·89-s + 2·95-s + 16·97-s + 6·101-s + 16·103-s + 2·109-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.80·11-s + 1.10·13-s + 1.45·17-s − 0.458·19-s + 1/5·25-s − 1.11·29-s + 1.79·31-s + 0.328·37-s − 0.937·41-s − 0.609·43-s + 1.64·53-s + 0.809·55-s − 1.79·61-s − 0.496·65-s − 0.488·67-s − 0.712·71-s + 0.468·73-s − 1.80·79-s − 1.31·83-s − 0.650·85-s + 0.635·89-s + 0.205·95-s + 1.62·97-s + 0.597·101-s + 1.57·103-s + 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.508427771\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.508427771\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−7.71992550315573670660381842088, −7.32203917067248011160417788772, −6.24499931726280904626653515463, −5.71243791878624412487897766878, −5.01671683693306102059151973202, −4.25325358586943313423876910079, −3.33486467081575469325328589432, −2.83499238967819126978690855366, −1.71100077734082612562550725372, −0.59636414169111811084024326572,
0.59636414169111811084024326572, 1.71100077734082612562550725372, 2.83499238967819126978690855366, 3.33486467081575469325328589432, 4.25325358586943313423876910079, 5.01671683693306102059151973202, 5.71243791878624412487897766878, 6.24499931726280904626653515463, 7.32203917067248011160417788772, 7.71992550315573670660381842088
