L(s) = 1 | − 2·5-s − 4·11-s − 6·13-s + 2·17-s − 4·19-s + 8·23-s − 25-s + 2·29-s − 10·37-s − 6·41-s + 4·43-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 + 4·83-s − 4·85-s − 6·89-s + 8·95-s + 14·97-s − 2·101-s + 8·103-s + 12·107-s + ⋯ |
L(s) = 1 | − 0.894·5-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 − 1.64·37-s − 0.937·41-s + 0.609·43-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 + 0.439·83-s − 0.433·85-s − 0.635·89-s + 0.820·95-s + 1.42·97-s − 0.199·101-s + 0.788·103-s + 1.16·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6882534522\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6882534522\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
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
−7.80482566859023040657170709303, −7.35990014451951924962892403121, −6.78096076788874774298742993601, −5.71112834827052213288460713965, −4.89765783876193378617287021019, −4.61184535669995648177321109940, −3.42407012452672903343907771151, −2.84079499225778567119125782528, −1.91012367697335827208229040334, −0.39903622293453192344515596766,
0.39903622293453192344515596766, 1.91012367697335827208229040334, 2.84079499225778567119125782528, 3.42407012452672903343907771151, 4.61184535669995648177321109940, 4.89765783876193378617287021019, 5.71112834827052213288460713965, 6.78096076788874774298742993601, 7.35990014451951924962892403121, 7.80482566859023040657170709303