| L(s) = 1 | + 5·5-s + 22.4·7-s − 56.5·11-s + 43.5·13-s − 34.9·17-s + 77.1·19-s − 122.·23-s + 25·25-s − 273.·29-s − 297.·31-s + 112.·35-s − 267.·37-s + 181.·41-s + 369.·43-s + 112.·47-s + 159.·49-s + 23.1·53-s − 282.·55-s − 279.·59-s − 392.·61-s + 217.·65-s + 394.·67-s − 973.·71-s − 760.·73-s − 1.26e3·77-s − 831.·79-s + 519.·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.21·7-s − 1.54·11-s + 0.929·13-s − 0.499·17-s + 0.931·19-s − 1.11·23-s + 0.200·25-s − 1.74·29-s − 1.72·31-s + 0.541·35-s − 1.19·37-s + 0.689·41-s + 1.31·43-s + 0.349·47-s + 0.466·49-s + 0.0599·53-s − 0.692·55-s − 0.616·59-s − 0.824·61-s + 0.415·65-s + 0.719·67-s − 1.62·71-s − 1.21·73-s − 1.87·77-s − 1.18·79-s + 0.687·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5T \) |
| good | 7 | \( 1 - 22.4T + 343T^{2} \) |
| 11 | \( 1 + 56.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 43.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 34.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 77.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 122.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 273.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 297.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 267.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 181.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 369.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 112.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 23.1T + 1.48e5T^{2} \) |
| 59 | \( 1 + 279.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 392.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 394.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 973.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 760.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 831.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 519.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.18e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 839.T + 9.12e5T^{2} \) |
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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
−8.648563033239192201794497950159, −7.69890093566613218375248679350, −7.37125201401777774758032679149, −5.82378265805412900538457831919, −5.52741817404935715709827462372, −4.54788729769674953789466895758, −3.50032888442676774938496527174, −2.25536407008940995601195610378, −1.51649404831975324363449201515, 0,
1.51649404831975324363449201515, 2.25536407008940995601195610378, 3.50032888442676774938496527174, 4.54788729769674953789466895758, 5.52741817404935715709827462372, 5.82378265805412900538457831919, 7.37125201401777774758032679149, 7.69890093566613218375248679350, 8.648563033239192201794497950159
