| L(s) = 1 | + 1.93·3-s + 3.94·7-s − 23.2·9-s − 29.7·11-s + 78.5·13-s + 53.0·17-s − 112.·19-s + 7.63·21-s − 114.·23-s − 97.2·27-s + 205.·29-s + 191.·31-s − 57.5·33-s − 365.·37-s + 152.·39-s − 407.·41-s + 293.·43-s + 155.·47-s − 327.·49-s + 102.·51-s − 309.·53-s − 217.·57-s − 620.·59-s + 437.·61-s − 91.8·63-s − 971.·67-s − 221.·69-s + ⋯ |
| L(s) = 1 | + 0.372·3-s + 0.213·7-s − 0.861·9-s − 0.815·11-s + 1.67·13-s + 0.756·17-s − 1.35·19-s + 0.0793·21-s − 1.03·23-s − 0.693·27-s + 1.31·29-s + 1.10·31-s − 0.303·33-s − 1.62·37-s + 0.624·39-s − 1.55·41-s + 1.03·43-s + 0.482·47-s − 0.954·49-s + 0.281·51-s − 0.803·53-s − 0.506·57-s − 1.36·59-s + 0.918·61-s − 0.183·63-s − 1.77·67-s − 0.386·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 1.93T + 27T^{2} \) |
| 7 | \( 1 - 3.94T + 343T^{2} \) |
| 11 | \( 1 + 29.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 78.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 53.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 112.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 114.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 205.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 191.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 365.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 407.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 293.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 155.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 309.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 620.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 437.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 971.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 508.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 560.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 198.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 327.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 259.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 548.T + 9.12e5T^{2} \) |
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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
−8.849472630439040540535117066415, −8.417076743753287360333875157519, −7.80904014852915548964551478197, −6.42819216421665948707295950591, −5.86750945999284194359619805620, −4.75743824744091651198149183204, −3.63806977179924991752844423910, −2.75774923494865100908767511651, −1.53388876282182382998154093456, 0,
1.53388876282182382998154093456, 2.75774923494865100908767511651, 3.63806977179924991752844423910, 4.75743824744091651198149183204, 5.86750945999284194359619805620, 6.42819216421665948707295950591, 7.80904014852915548964551478197, 8.417076743753287360333875157519, 8.849472630439040540535117066415
