L(s) = 1 | − 2·3-s + 9-s − 3·11-s + 13-s − 2·17-s − 19-s − 23-s + 4·27-s + 2·29-s + 4·31-s + 6·33-s + 9·37-s − 2·39-s − 3·41-s − 2·43-s + 9·47-s + 4·51-s + 9·53-s + 2·57-s + 12·61-s + 8·67-s + 2·69-s + 16·71-s + 4·73-s + 6·79-s − 11·81-s + 4·83-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1/3·9-s − 0.904·11-s + 0.277·13-s − 0.485·17-s − 0.229·19-s − 0.208·23-s + 0.769·27-s + 0.371·29-s + 0.718·31-s + 1.04·33-s + 1.47·37-s − 0.320·39-s − 0.468·41-s − 0.304·43-s + 1.31·47-s + 0.560·51-s + 1.23·53-s + 0.264·57-s + 1.53·61-s + 0.977·67-s + 0.240·69-s + 1.89·71-s + 0.468·73-s + 0.675·79-s − 1.22·81-s + 0.439·83-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
\(L(1)\) |
\(\approx\) |
\(1.481768569\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.481768569\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.99469066418823, −13.41567276653804, −12.91243045279543, −12.56199470232183, −11.87896198609209, −11.55979328432317, −11.01097219273224, −10.66004067045902, −10.08071774845850, −9.728963743321862, −8.854817564551733, −8.442229945236183, −7.916958298412203, −7.267924452269349, −6.661018121963036, −6.226607131174074, −5.729749564183310, −5.141750136949726, −4.755629258107109, −4.075781920083997, −3.413534264865947, −2.493203028720906, −2.177579317153030, −0.9043989423388748, −0.5595653790660109,
0.5595653790660109, 0.9043989423388748, 2.177579317153030, 2.493203028720906, 3.413534264865947, 4.075781920083997, 4.755629258107109, 5.141750136949726, 5.729749564183310, 6.226607131174074, 6.661018121963036, 7.267924452269349, 7.916958298412203, 8.442229945236183, 8.854817564551733, 9.728963743321862, 10.08071774845850, 10.66004067045902, 11.01097219273224, 11.55979328432317, 11.87896198609209, 12.56199470232183, 12.91243045279543, 13.41567276653804, 13.99469066418823