L(s) = 1 | − 5-s + 2·7-s − 6·13-s + 7·17-s − 7·19-s − 7·23-s + 25-s + 6·29-s − 3·31-s − 2·35-s − 6·37-s + 4·41-s − 8·43-s + 4·47-s − 3·49-s − 5·53-s − 6·59-s − 3·61-s + 6·65-s + 10·67-s − 12·71-s + 16·73-s − 79-s − 9·83-s − 7·85-s − 4·89-s − 12·91-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.755·7-s − 1.66·13-s + 1.69·17-s − 1.60·19-s − 1.45·23-s + 1/5·25-s + 1.11·29-s − 0.538·31-s − 0.338·35-s − 0.986·37-s + 0.624·41-s − 1.21·43-s + 0.583·47-s − 3/7·49-s − 0.686·53-s − 0.781·59-s − 0.384·61-s + 0.744·65-s + 1.22·67-s − 1.42·71-s + 1.87·73-s − 0.112·79-s − 0.987·83-s − 0.759·85-s − 0.423·89-s − 1.25·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−8.417641292312177040514966117714, −7.987877301364546447873215900931, −7.29660178743434381181891232968, −6.37517275868954027650164059326, −5.35222972072874174258737836538, −4.67846855320387702510100298813, −3.83307707888875378370578455990, −2.67825442003368599080861022128, −1.65663735234511009393828811945, 0,
1.65663735234511009393828811945, 2.67825442003368599080861022128, 3.83307707888875378370578455990, 4.67846855320387702510100298813, 5.35222972072874174258737836538, 6.37517275868954027650164059326, 7.29660178743434381181891232968, 7.987877301364546447873215900931, 8.417641292312177040514966117714