L(s) = 1 | + 2·5-s + 4·7-s − 6·13-s + 6·17-s − 8·19-s − 25-s − 6·29-s + 8·35-s + 6·37-s − 10·41-s − 8·43-s + 9·49-s − 6·53-s + 4·59-s + 2·61-s − 12·65-s + 12·67-s − 8·71-s − 2·73-s − 4·79-s + 12·83-s + 12·85-s + 6·89-s − 24·91-s − 16·95-s + 2·97-s + 101-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 1.51·7-s − 1.66·13-s + 1.45·17-s − 1.83·19-s − 1/5·25-s − 1.11·29-s + 1.35·35-s + 0.986·37-s − 1.56·41-s − 1.21·43-s + 9/7·49-s − 0.824·53-s + 0.520·59-s + 0.256·61-s − 1.48·65-s + 1.46·67-s − 0.949·71-s − 0.234·73-s − 0.450·79-s + 1.31·83-s + 1.30·85-s + 0.635·89-s − 2.51·91-s − 1.64·95-s + 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17424 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−16.36919347050276, −15.23314572569715, −14.81097742985787, −14.62791225472520, −14.09908903681726, −13.35769298296746, −12.85272040668667, −12.20142058046745, −11.69576637667248, −11.15210402493362, −10.35170860319082, −10.04893382159807, −9.450922567286075, −8.727933511872652, −7.971488754196930, −7.764293523412562, −6.902734681609384, −6.235915256397293, −5.382931676298422, −5.116715206704746, −4.440614245486270, −3.610240274551057, −2.501922905820217, −2.001616135373072, −1.389624160696307, 0,
1.389624160696307, 2.001616135373072, 2.501922905820217, 3.610240274551057, 4.440614245486270, 5.116715206704746, 5.382931676298422, 6.235915256397293, 6.902734681609384, 7.764293523412562, 7.971488754196930, 8.727933511872652, 9.450922567286075, 10.04893382159807, 10.35170860319082, 11.15210402493362, 11.69576637667248, 12.20142058046745, 12.85272040668667, 13.35769298296746, 14.09908903681726, 14.62791225472520, 14.81097742985787, 15.23314572569715, 16.36919347050276