| L(s) = 1 | + 5-s − 7-s − 2·13-s + 2·19-s + 25-s − 6·29-s − 8·31-s − 35-s + 4·37-s + 6·41-s + 2·43-s + 6·47-s + 49-s − 6·53-s + 12·59-s − 8·61-s − 2·65-s + 2·67-s − 6·71-s + 2·73-s + 16·79-s + 6·89-s + 2·91-s + 2·95-s − 10·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 0.554·13-s + 0.458·19-s + 1/5·25-s − 1.11·29-s − 1.43·31-s − 0.169·35-s + 0.657·37-s + 0.937·41-s + 0.304·43-s + 0.875·47-s + 1/7·49-s − 0.824·53-s + 1.56·59-s − 1.02·61-s − 0.248·65-s + 0.244·67-s − 0.712·71-s + 0.234·73-s + 1.80·79-s + 0.635·89-s + 0.209·91-s + 0.205·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−16.16825252497289, −15.18956568267528, −14.92030850656621, −14.27559780426255, −13.75038223729267, −13.19633284970678, −12.57697134085764, −12.32645159203302, −11.35938121594861, −11.05832914599957, −10.32458691509848, −9.741307001386054, −9.237426147857335, −8.882633283449104, −7.820351716458529, −7.503742324959007, −6.820442778269800, −6.110126617693985, −5.550724980195710, −5.015649734201103, −4.116427860511812, −3.535331803084809, −2.662233939479869, −2.067524098218160, −1.099884647544111, 0,
1.099884647544111, 2.067524098218160, 2.662233939479869, 3.535331803084809, 4.116427860511812, 5.015649734201103, 5.550724980195710, 6.110126617693985, 6.820442778269800, 7.503742324959007, 7.820351716458529, 8.882633283449104, 9.237426147857335, 9.741307001386054, 10.32458691509848, 11.05832914599957, 11.35938121594861, 12.32645159203302, 12.57697134085764, 13.19633284970678, 13.75038223729267, 14.27559780426255, 14.92030850656621, 15.18956568267528, 16.16825252497289
