| L(s) = 1 | − 1.22·3-s + 0.177·5-s + 3.50·7-s − 1.49·9-s + 2.72·11-s − 5.04·13-s − 0.217·15-s − 7.46·17-s − 8.22·19-s − 4.29·21-s − 4.99·23-s − 4.96·25-s + 5.51·27-s + 8.04·29-s − 6.68·31-s − 3.34·33-s + 0.623·35-s + 10.9·37-s + 6.17·39-s − 1.44·41-s + 8.34·43-s − 0.266·45-s + 47-s + 5.29·49-s + 9.14·51-s + 0.281·53-s + 0.485·55-s + ⋯ |
| L(s) = 1 | − 0.707·3-s + 0.0794·5-s + 1.32·7-s − 0.499·9-s + 0.822·11-s − 1.39·13-s − 0.0562·15-s − 1.80·17-s − 1.88·19-s − 0.937·21-s − 1.04·23-s − 0.993·25-s + 1.06·27-s + 1.49·29-s − 1.20·31-s − 0.582·33-s + 0.105·35-s + 1.79·37-s + 0.989·39-s − 0.225·41-s + 1.27·43-s − 0.0396·45-s + 0.145·47-s + 0.756·49-s + 1.28·51-s + 0.0386·53-s + 0.0654·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.104152160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.104152160\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 3 | \( 1 + 1.22T + 3T^{2} \) |
| 5 | \( 1 - 0.177T + 5T^{2} \) |
| 7 | \( 1 - 3.50T + 7T^{2} \) |
| 11 | \( 1 - 2.72T + 11T^{2} \) |
| 13 | \( 1 + 5.04T + 13T^{2} \) |
| 17 | \( 1 + 7.46T + 17T^{2} \) |
| 19 | \( 1 + 8.22T + 19T^{2} \) |
| 23 | \( 1 + 4.99T + 23T^{2} \) |
| 29 | \( 1 - 8.04T + 29T^{2} \) |
| 31 | \( 1 + 6.68T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 1.44T + 41T^{2} \) |
| 43 | \( 1 - 8.34T + 43T^{2} \) |
| 53 | \( 1 - 0.281T + 53T^{2} \) |
| 59 | \( 1 - 7.20T + 59T^{2} \) |
| 61 | \( 1 - 3.44T + 61T^{2} \) |
| 67 | \( 1 - 4.06T + 67T^{2} \) |
| 71 | \( 1 - 2.32T + 71T^{2} \) |
| 73 | \( 1 - 3.81T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 16.3T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−8.137737531877569799235963482502, −7.36001035729701574044799607954, −6.43146425857680175180667878040, −6.11375568037464100164933747406, −5.08453515970225728987390353179, −4.50695138633113884500289589170, −4.03709120320317826580041446305, −2.27716411358178050450657557418, −2.13804322901129505026053206264, −0.54661733900152232827890762735,
0.54661733900152232827890762735, 2.13804322901129505026053206264, 2.27716411358178050450657557418, 4.03709120320317826580041446305, 4.50695138633113884500289589170, 5.08453515970225728987390353179, 6.11375568037464100164933747406, 6.43146425857680175180667878040, 7.36001035729701574044799607954, 8.137737531877569799235963482502
