L(s) = 1 | + 2·5-s + 7-s − 4·11-s + 6·13-s − 2·17-s + 4·19-s + 8·23-s − 25-s + 2·29-s + 2·35-s − 10·37-s + 6·41-s + 4·43-s + 49-s − 6·53-s − 8·55-s + 4·59-s + 6·61-s + 12·65-s − 4·67-s + 8·71-s + 10·73-s − 4·77-s − 4·83-s − 4·85-s + 6·89-s + 6·91-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 0.377·7-s − 1.20·11-s + 1.66·13-s − 0.485·17-s + 0.917·19-s + 1.66·23-s − 1/5·25-s + 0.371·29-s + 0.338·35-s − 1.64·37-s + 0.937·41-s + 0.609·43-s + 1/7·49-s − 0.824·53-s − 1.07·55-s + 0.520·59-s + 0.768·61-s + 1.48·65-s − 0.488·67-s + 0.949·71-s + 1.17·73-s − 0.455·77-s − 0.439·83-s − 0.433·85-s + 0.635·89-s + 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.006550525\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.006550525\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−9.998713269577231931539325852863, −9.085422624242741773351689713512, −8.427817591055194605299416158333, −7.46998199559376075566652373057, −6.48878700424638503015501865218, −5.58110052520146908580921209772, −4.96240850233932644050318806427, −3.57539886636491016875562737772, −2.47676327970158091646809943718, −1.22168166913217334012046287130,
1.22168166913217334012046287130, 2.47676327970158091646809943718, 3.57539886636491016875562737772, 4.96240850233932644050318806427, 5.58110052520146908580921209772, 6.48878700424638503015501865218, 7.46998199559376075566652373057, 8.427817591055194605299416158333, 9.085422624242741773351689713512, 9.998713269577231931539325852863