L(s) = 1 | + 4-s − 1.80·5-s + 9-s − 0.445·13-s + 16-s − 1.80·20-s + 1.24·23-s + 2.24·25-s + 1.24·31-s + 36-s − 0.445·41-s − 0.445·43-s − 1.80·45-s + 49-s − 0.445·52-s + 64-s + 0.801·65-s − 0.445·71-s − 1.80·80-s + 81-s + 1.24·83-s + 1.24·89-s + 1.24·92-s + 2.24·100-s − 1.80·101-s − 1.80·103-s − 1.80·109-s + ⋯ |
L(s) = 1 | + 4-s − 1.80·5-s + 9-s − 0.445·13-s + 16-s − 1.80·20-s + 1.24·23-s + 2.24·25-s + 1.24·31-s + 36-s − 0.445·41-s − 0.445·43-s − 1.80·45-s + 49-s − 0.445·52-s + 64-s + 0.801·65-s − 0.445·71-s − 1.80·80-s + 81-s + 1.24·83-s + 1.24·89-s + 1.24·92-s + 2.24·100-s − 1.80·101-s − 1.80·103-s − 1.80·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.179929084\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.179929084\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2011 | \( 1+O(T) \) |
good | 2 | \( 1 - T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 + 1.80T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 0.445T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 1.24T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 1.24T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 0.445T + T^{2} \) |
| 43 | \( 1 + 0.445T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + 0.445T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 1.24T + T^{2} \) |
| 89 | \( 1 - 1.24T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.307637544819940100064989679548, −8.260296202951996999396743795541, −7.71034924230206571734003232675, −7.03384206160308353072004579797, −6.60257240899894917415705156396, −5.17284836000195726489118549613, −4.34831431540597383193760842166, −3.52165873841477642791745100310, −2.64131041767919126497260176640, −1.14161997012900792784024894394,
1.14161997012900792784024894394, 2.64131041767919126497260176640, 3.52165873841477642791745100310, 4.34831431540597383193760842166, 5.17284836000195726489118549613, 6.60257240899894917415705156396, 7.03384206160308353072004579797, 7.71034924230206571734003232675, 8.260296202951996999396743795541, 9.307637544819940100064989679548