L(s) = 1 | + 2·5-s − 7-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 + 4·59-s − 6·61-s − 12·65-s − 4·67-s + 8·71-s − 10·73-s + 4·83-s + 4·85-s + 6·89-s + 6·91-s − 8·95-s − 14·97-s + 101-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 0.377·7-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 + 0.520·59-s − 0.768·61-s − 1.48·65-s − 0.488·67-s + 0.949·71-s − 1.17·73-s + 0.439·83-s + 0.433·85-s + 0.635·89-s + 0.628·91-s − 0.820·95-s − 1.42·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.098072642\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.098072642\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 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
−13.55823153859254, −13.07402418756941, −12.57724364295478, −12.22651152768272, −11.73033531898999, −11.02913478738343, −10.50993018602408, −10.15905661028774, −9.565599685688184, −9.370862150068550, −8.683798700609256, −8.233725842664781, −7.430475855501263, −7.084511780615503, −6.614747338039487, −6.049329566126219, −5.350505256925810, −5.073752917202162, −4.536410810423406, −3.691443618828162, −3.097086424150100, −2.571101477573349, −1.923365803793868, −1.405975164439940, −0.3016747969697490,
0.3016747969697490, 1.405975164439940, 1.923365803793868, 2.571101477573349, 3.097086424150100, 3.691443618828162, 4.536410810423406, 5.073752917202162, 5.350505256925810, 6.049329566126219, 6.614747338039487, 7.084511780615503, 7.430475855501263, 8.233725842664781, 8.683798700609256, 9.370862150068550, 9.565599685688184, 10.15905661028774, 10.50993018602408, 11.02913478738343, 11.73033531898999, 12.22651152768272, 12.57724364295478, 13.07402418756941, 13.55823153859254