| L(s) = 1 | + 7-s − 3·9-s − 4·11-s − 2·13-s + 6·17-s + 8·19-s + 6·29-s + 8·31-s + 2·37-s + 2·41-s + 4·43-s + 8·47-s + 49-s − 6·53-s − 6·61-s − 3·63-s + 4·67-s − 8·71-s − 10·73-s − 4·77-s + 16·79-s + 9·81-s − 8·83-s − 6·89-s − 2·91-s + 6·97-s + 12·99-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 9-s − 1.20·11-s − 0.554·13-s + 1.45·17-s + 1.83·19-s + 1.11·29-s + 1.43·31-s + 0.328·37-s + 0.312·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.824·53-s − 0.768·61-s − 0.377·63-s + 0.488·67-s − 0.949·71-s − 1.17·73-s − 0.455·77-s + 1.80·79-s + 81-s − 0.878·83-s − 0.635·89-s − 0.209·91-s + 0.609·97-s + 1.20·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.564439584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.564439584\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 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 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−9.694440338392361408863185872440, −8.660773281175179322159496718137, −7.81362030580797675551267597140, −7.47489161542833731239859573103, −6.06292563014554710285489032478, −5.38565578310165680979646788554, −4.71019542661529810418780789156, −3.18975677352219866037498641634, −2.62755636242588070812542768623, −0.916171764469637391994006820348,
0.916171764469637391994006820348, 2.62755636242588070812542768623, 3.18975677352219866037498641634, 4.71019542661529810418780789156, 5.38565578310165680979646788554, 6.06292563014554710285489032478, 7.47489161542833731239859573103, 7.81362030580797675551267597140, 8.660773281175179322159496718137, 9.694440338392361408863185872440
