L(s) = 1 | + 5-s − 2·7-s − 4·13-s − 2·17-s − 4·19-s − 8·23-s + 25-s − 10·29-s + 4·31-s − 2·35-s + 8·43-s − 8·47-s − 3·49-s − 6·53-s + 14·59-s + 14·61-s − 4·65-s − 4·67-s − 12·71-s − 6·73-s + 12·79-s + 4·83-s − 2·85-s + 12·89-s + 8·91-s − 4·95-s − 14·97-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.755·7-s − 1.10·13-s − 0.485·17-s − 0.917·19-s − 1.66·23-s + 1/5·25-s − 1.85·29-s + 0.718·31-s − 0.338·35-s + 1.21·43-s − 1.16·47-s − 3/7·49-s − 0.824·53-s + 1.82·59-s + 1.79·61-s − 0.496·65-s − 0.488·67-s − 1.42·71-s − 0.702·73-s + 1.35·79-s + 0.439·83-s − 0.216·85-s + 1.27·89-s + 0.838·91-s − 0.410·95-s − 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4993120978\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4993120978\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−14.64144303811192, −14.31404368211007, −13.51058057977360, −13.07820097175093, −12.78811005585312, −12.10239581687416, −11.65299334330258, −11.02951252177860, −10.28309745615702, −10.06326884136871, −9.331535017591498, −9.176817883619970, −8.172266827163266, −7.881871044547400, −7.056027886702727, −6.597079438043118, −6.071440710963655, −5.492563192990125, −4.856132507443906, −4.070184732653736, −3.695762129261614, −2.629207621715733, −2.302569094189798, −1.538492227839355, −0.2386648175201205,
0.2386648175201205, 1.538492227839355, 2.302569094189798, 2.629207621715733, 3.695762129261614, 4.070184732653736, 4.856132507443906, 5.492563192990125, 6.071440710963655, 6.597079438043118, 7.056027886702727, 7.881871044547400, 8.172266827163266, 9.176817883619970, 9.331535017591498, 10.06326884136871, 10.28309745615702, 11.02951252177860, 11.65299334330258, 12.10239581687416, 12.78811005585312, 13.07820097175093, 13.51058057977360, 14.31404368211007, 14.64144303811192