| L(s) = 1 | − 2.44·3-s − 5-s − 2.44·7-s + 2.99·9-s − 5.09·11-s + 4·13-s + 2.44·15-s − 5.24·17-s − 2.44·19-s + 5.99·21-s − 5.09·23-s + 25-s + 29-s − 7.34·31-s + 12.4·33-s + 2.44·35-s − 9.24·37-s − 9.79·39-s − 8·41-s + 7.74·43-s − 2.99·45-s − 12.6·47-s − 1.00·49-s + 12.8·51-s + 8·53-s + 5.09·55-s + 5.99·57-s + ⋯ |
| L(s) = 1 | − 1.41·3-s − 0.447·5-s − 0.925·7-s + 0.999·9-s − 1.53·11-s + 1.10·13-s + 0.632·15-s − 1.27·17-s − 0.561·19-s + 1.30·21-s − 1.06·23-s + 0.200·25-s + 0.185·29-s − 1.31·31-s + 2.17·33-s + 0.414·35-s − 1.51·37-s − 1.56·39-s − 1.24·41-s + 1.18·43-s − 0.447·45-s − 1.84·47-s − 0.142·49-s + 1.79·51-s + 1.09·53-s + 0.687·55-s + 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.03660113934\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03660113934\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.44T + 3T^{2} \) |
| 7 | \( 1 + 2.44T + 7T^{2} \) |
| 11 | \( 1 + 5.09T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 5.24T + 17T^{2} \) |
| 19 | \( 1 + 2.44T + 19T^{2} \) |
| 23 | \( 1 + 5.09T + 23T^{2} \) |
| 31 | \( 1 + 7.34T + 31T^{2} \) |
| 37 | \( 1 + 9.24T + 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 - 7.74T + 43T^{2} \) |
| 47 | \( 1 + 12.6T + 47T^{2} \) |
| 53 | \( 1 - 8T + 53T^{2} \) |
| 59 | \( 1 - 2.24T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 - 0.200T + 67T^{2} \) |
| 71 | \( 1 - 2.64T + 71T^{2} \) |
| 73 | \( 1 + 0.755T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 + 17.5T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−8.380286824567959620864190915325, −7.37871203026847040134617378932, −6.69901432456734961352531897293, −6.10190458079621508203200183727, −5.50152090957215464113928057035, −4.72495961665427389426741301350, −3.89475352544311362113058912629, −2.99239123183103189927885208403, −1.77870583718928302213114608961, −0.10875817284658066471889945186,
0.10875817284658066471889945186, 1.77870583718928302213114608961, 2.99239123183103189927885208403, 3.89475352544311362113058912629, 4.72495961665427389426741301350, 5.50152090957215464113928057035, 6.10190458079621508203200183727, 6.69901432456734961352531897293, 7.37871203026847040134617378932, 8.380286824567959620864190915325
