L(s) = 1 | − 3·3-s − 5-s + 6·9-s − 11-s + 3·13-s + 3·15-s − 3·17-s − 6·19-s + 4·23-s + 25-s − 9·27-s − 29-s − 6·31-s + 3·33-s − 9·39-s + 6·41-s + 6·43-s − 6·45-s + 9·47-s + 9·51-s − 10·53-s + 55-s + 18·57-s + 6·59-s − 3·65-s + 14·67-s − 12·69-s + ⋯ |
L(s) = 1 | − 1.73·3-s − 0.447·5-s + 2·9-s − 0.301·11-s + 0.832·13-s + 0.774·15-s − 0.727·17-s − 1.37·19-s + 0.834·23-s + 1/5·25-s − 1.73·27-s − 0.185·29-s − 1.07·31-s + 0.522·33-s − 1.44·39-s + 0.937·41-s + 0.914·43-s − 0.894·45-s + 1.31·47-s + 1.26·51-s − 1.37·53-s + 0.134·55-s + 2.38·57-s + 0.781·59-s − 0.372·65-s + 1.71·67-s − 1.44·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 15 T + p T^{2} \) |
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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
−8.005554576855030506973395247744, −7.08805432499483962275697666178, −6.57091628183451963548729368663, −5.86533015498480812665107502973, −5.20845235821363696404067822215, −4.37721416988047421172244041901, −3.79401301888147171856189613274, −2.31495449539805993845775020354, −1.04857418577485268402668080063, 0,
1.04857418577485268402668080063, 2.31495449539805993845775020354, 3.79401301888147171856189613274, 4.37721416988047421172244041901, 5.20845235821363696404067822215, 5.86533015498480812665107502973, 6.57091628183451963548729368663, 7.08805432499483962275697666178, 8.005554576855030506973395247744