| L(s) = 1 | + 3-s + 2.37·5-s + 7-s − 2·9-s − 4.37·11-s − 2.37·13-s + 2.37·15-s + 2·17-s − 7.37·19-s + 21-s − 5.37·23-s + 0.627·25-s − 5·27-s − 29-s + 7.11·31-s − 4.37·33-s + 2.37·35-s − 4·37-s − 2.37·39-s + 0.627·41-s − 8.37·43-s − 4.74·45-s − 9.74·47-s + 49-s + 2·51-s + 9.74·53-s − 10.3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.06·5-s + 0.377·7-s − 0.666·9-s − 1.31·11-s − 0.657·13-s + 0.612·15-s + 0.485·17-s − 1.69·19-s + 0.218·21-s − 1.12·23-s + 0.125·25-s − 0.962·27-s − 0.185·29-s + 1.27·31-s − 0.761·33-s + 0.400·35-s − 0.657·37-s − 0.379·39-s + 0.0980·41-s − 1.27·43-s − 0.707·45-s − 1.42·47-s + 0.142·49-s + 0.280·51-s + 1.33·53-s − 1.39·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - T + 3T^{2} \) |
| 5 | \( 1 - 2.37T + 5T^{2} \) |
| 11 | \( 1 + 4.37T + 11T^{2} \) |
| 13 | \( 1 + 2.37T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 7.37T + 19T^{2} \) |
| 23 | \( 1 + 5.37T + 23T^{2} \) |
| 31 | \( 1 - 7.11T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 0.627T + 41T^{2} \) |
| 43 | \( 1 + 8.37T + 43T^{2} \) |
| 47 | \( 1 + 9.74T + 47T^{2} \) |
| 53 | \( 1 - 9.74T + 53T^{2} \) |
| 59 | \( 1 + 8.74T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 7.37T + 67T^{2} \) |
| 71 | \( 1 - 5.37T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 + 10T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + 0.627T + 97T^{2} \) |
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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
−8.167602759348633356152768676842, −7.896353527514638694523416005693, −6.69394246064281083795720330422, −5.93769898924469230420339723574, −5.29112941211912007289303150083, −4.50580771573554477325171553715, −3.28491907249612908477283635918, −2.37024763087187552478117523093, −1.92712589790890461493961451724, 0,
1.92712589790890461493961451724, 2.37024763087187552478117523093, 3.28491907249612908477283635918, 4.50580771573554477325171553715, 5.29112941211912007289303150083, 5.93769898924469230420339723574, 6.69394246064281083795720330422, 7.896353527514638694523416005693, 8.167602759348633356152768676842
