| L(s) = 1 | − 0.462·3-s + 5-s + 2.86·7-s − 2.78·9-s + 5.72·11-s − 0.462·13-s − 0.462·15-s + 1.93·17-s − 8.36·19-s − 1.32·21-s − 8.18·23-s + 25-s + 2.67·27-s + 29-s − 3.78·31-s − 2.64·33-s + 2.86·35-s − 3.07·37-s + 0.213·39-s + 3.72·41-s + 9.78·43-s − 2.78·45-s − 8.64·47-s + 1.18·49-s − 0.895·51-s + 2.58·53-s + 5.72·55-s + ⋯ |
| L(s) = 1 | − 0.267·3-s + 0.447·5-s + 1.08·7-s − 0.928·9-s + 1.72·11-s − 0.128·13-s − 0.119·15-s + 0.469·17-s − 1.91·19-s − 0.288·21-s − 1.70·23-s + 0.200·25-s + 0.515·27-s + 0.185·29-s − 0.679·31-s − 0.460·33-s + 0.483·35-s − 0.505·37-s + 0.0342·39-s + 0.581·41-s + 1.49·43-s − 0.415·45-s − 1.26·47-s + 0.169·49-s − 0.125·51-s + 0.354·53-s + 0.771·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.230878205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.230878205\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 0.462T + 3T^{2} \) |
| 7 | \( 1 - 2.86T + 7T^{2} \) |
| 11 | \( 1 - 5.72T + 11T^{2} \) |
| 13 | \( 1 + 0.462T + 13T^{2} \) |
| 17 | \( 1 - 1.93T + 17T^{2} \) |
| 19 | \( 1 + 8.36T + 19T^{2} \) |
| 23 | \( 1 + 8.18T + 23T^{2} \) |
| 31 | \( 1 + 3.78T + 31T^{2} \) |
| 37 | \( 1 + 3.07T + 37T^{2} \) |
| 41 | \( 1 - 3.72T + 41T^{2} \) |
| 43 | \( 1 - 9.78T + 43T^{2} \) |
| 47 | \( 1 + 8.64T + 47T^{2} \) |
| 53 | \( 1 - 2.58T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 - 9.62T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 - 5.29T + 71T^{2} \) |
| 73 | \( 1 - 9.63T + 73T^{2} \) |
| 79 | \( 1 - 5.90T + 79T^{2} \) |
| 83 | \( 1 - 4.92T + 83T^{2} \) |
| 89 | \( 1 + 5.07T + 89T^{2} \) |
| 97 | \( 1 + 1.66T + 97T^{2} \) |
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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
−7.906234730169660413735295633008, −6.79417013220734020787557846121, −6.34914789038921333918668724714, −5.71176227920688126252453378586, −5.03264095829103997512613364689, −4.13210513775735287392023834763, −3.70267136461340902513573728072, −2.28955734635908674896193149515, −1.87037622646664538335610037609, −0.73133404863813203341710337324,
0.73133404863813203341710337324, 1.87037622646664538335610037609, 2.28955734635908674896193149515, 3.70267136461340902513573728072, 4.13210513775735287392023834763, 5.03264095829103997512613364689, 5.71176227920688126252453378586, 6.34914789038921333918668724714, 6.79417013220734020787557846121, 7.906234730169660413735295633008
