L(s) = 1 | − 5-s − 2·11-s − 2·13-s − 6·17-s + 2·19-s + 4·23-s + 25-s + 8·29-s − 10·31-s − 10·37-s − 6·41-s + 8·43-s + 8·47-s − 14·53-s + 2·55-s + 4·59-s + 2·65-s + 4·67-s + 6·71-s − 10·73-s + 16·79-s − 4·83-s + 6·85-s − 10·89-s − 2·95-s + 10·97-s + 101-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.603·11-s − 0.554·13-s − 1.45·17-s + 0.458·19-s + 0.834·23-s + 1/5·25-s + 1.48·29-s − 1.79·31-s − 1.64·37-s − 0.937·41-s + 1.21·43-s + 1.16·47-s − 1.92·53-s + 0.269·55-s + 0.520·59-s + 0.248·65-s + 0.488·67-s + 0.712·71-s − 1.17·73-s + 1.80·79-s − 0.439·83-s + 0.650·85-s − 1.05·89-s − 0.205·95-s + 1.01·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−13.63923364946158, −13.13588594307762, −12.57933762091773, −12.33704198785530, −11.74964156671432, −11.14130426835107, −10.77066846407935, −10.49539133973851, −9.684938233783694, −9.323814093273297, −8.622869157457572, −8.500471726901912, −7.684492345052471, −7.191786617570796, −6.941296535862190, −6.285355389829165, −5.610236999045949, −4.955034514951740, −4.789646563587396, −4.018135555287657, −3.430538709834118, −2.853989508716563, −2.259567333646014, −1.639164275417006, −0.6819112651201299, 0,
0.6819112651201299, 1.639164275417006, 2.259567333646014, 2.853989508716563, 3.430538709834118, 4.018135555287657, 4.789646563587396, 4.955034514951740, 5.610236999045949, 6.285355389829165, 6.941296535862190, 7.191786617570796, 7.684492345052471, 8.500471726901912, 8.622869157457572, 9.323814093273297, 9.684938233783694, 10.49539133973851, 10.77066846407935, 11.14130426835107, 11.74964156671432, 12.33704198785530, 12.57933762091773, 13.13588594307762, 13.63923364946158