L(s) = 1 | − 3-s − 2·9-s − 2·11-s + 4·17-s − 2·19-s − 23-s + 5·27-s − 9·29-s − 4·31-s + 2·33-s + 4·37-s − 41-s + 9·43-s − 4·51-s − 10·53-s + 2·57-s − 10·59-s + 9·61-s + 5·67-s + 69-s + 14·71-s + 12·73-s + 14·79-s + 81-s − 11·83-s + 9·87-s + 15·89-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 2/3·9-s − 0.603·11-s + 0.970·17-s − 0.458·19-s − 0.208·23-s + 0.962·27-s − 1.67·29-s − 0.718·31-s + 0.348·33-s + 0.657·37-s − 0.156·41-s + 1.37·43-s − 0.560·51-s − 1.37·53-s + 0.264·57-s − 1.30·59-s + 1.15·61-s + 0.610·67-s + 0.120·69-s + 1.66·71-s + 1.40·73-s + 1.57·79-s + 1/9·81-s − 1.20·83-s + 0.964·87-s + 1.58·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9021495554\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9021495554\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−14.13727968081775, −13.49649214340604, −12.88824184418418, −12.46506213367149, −12.17212064785753, −11.32771646632643, −11.03685912549586, −10.76928347332439, −9.998387624279034, −9.478471202007795, −9.095394135691133, −8.327392140641612, −7.804065598807003, −7.561023054836964, −6.641369670210201, −6.263322593976194, −5.489897008515915, −5.409213774290939, −4.697132239308362, −3.852237074573940, −3.448786308949114, −2.611912621936091, −2.104393747162055, −1.176989751318968, −0.3424755097835835,
0.3424755097835835, 1.176989751318968, 2.104393747162055, 2.611912621936091, 3.448786308949114, 3.852237074573940, 4.697132239308362, 5.409213774290939, 5.489897008515915, 6.263322593976194, 6.641369670210201, 7.561023054836964, 7.804065598807003, 8.327392140641612, 9.095394135691133, 9.478471202007795, 9.998387624279034, 10.76928347332439, 11.03685912549586, 11.32771646632643, 12.17212064785753, 12.46506213367149, 12.88824184418418, 13.49649214340604, 14.13727968081775