| L(s) = 1 | + 5-s + 4·11-s − 4·13-s + 6·17-s + 6·19-s + 6·23-s + 25-s + 2·29-s − 8·31-s + 8·37-s + 4·43-s − 6·47-s − 7·49-s − 10·53-s + 4·55-s − 12·59-s + 6·61-s − 4·65-s − 8·67-s + 4·71-s + 14·73-s − 4·79-s − 4·83-s + 6·85-s + 4·89-s + 6·95-s + 18·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.20·11-s − 1.10·13-s + 1.45·17-s + 1.37·19-s + 1.25·23-s + 1/5·25-s + 0.371·29-s − 1.43·31-s + 1.31·37-s + 0.609·43-s − 0.875·47-s − 49-s − 1.37·53-s + 0.539·55-s − 1.56·59-s + 0.768·61-s − 0.496·65-s − 0.977·67-s + 0.474·71-s + 1.63·73-s − 0.450·79-s − 0.439·83-s + 0.650·85-s + 0.423·89-s + 0.615·95-s + 1.82·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.562990514\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.562990514\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−7.915498693344195473459991934156, −7.45645405407327466794189181251, −6.72715814240596862911710437882, −5.95657367615252029699134653277, −5.21208760644277380449490969485, −4.63080528810134641587934365750, −3.46642982637163098219628506995, −2.96149597344385008296549009236, −1.72613225983972960150385260557, −0.910382245506312098936969232690,
0.910382245506312098936969232690, 1.72613225983972960150385260557, 2.96149597344385008296549009236, 3.46642982637163098219628506995, 4.63080528810134641587934365750, 5.21208760644277380449490969485, 5.95657367615252029699134653277, 6.72715814240596862911710437882, 7.45645405407327466794189181251, 7.915498693344195473459991934156
