L(s) = 1 | − 3-s − 5-s − 2·9-s − 2·11-s + 4·13-s + 15-s + 6·19-s + 3·23-s + 25-s + 5·27-s − 3·29-s + 2·33-s − 12·37-s − 4·39-s − 7·41-s − 9·43-s + 2·45-s − 6·53-s + 2·55-s − 6·57-s − 10·59-s + 5·61-s − 4·65-s + 11·67-s − 3·69-s − 10·71-s − 8·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 2/3·9-s − 0.603·11-s + 1.10·13-s + 0.258·15-s + 1.37·19-s + 0.625·23-s + 1/5·25-s + 0.962·27-s − 0.557·29-s + 0.348·33-s − 1.97·37-s − 0.640·39-s − 1.09·41-s − 1.37·43-s + 0.298·45-s − 0.824·53-s + 0.269·55-s − 0.794·57-s − 1.30·59-s + 0.640·61-s − 0.496·65-s + 1.34·67-s − 0.361·69-s − 1.18·71-s − 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 17 T + p T^{2} \) |
| 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
−8.664044103642081964361887443345, −8.106170928651121440047566664440, −7.15583529024822094542501370686, −6.40713018566319065782761915503, −5.41340289830795980497616333727, −5.01968135180339698653294960296, −3.63504624826388638450776523592, −3.01165179487227605482580787317, −1.42716328319966679433068843180, 0,
1.42716328319966679433068843180, 3.01165179487227605482580787317, 3.63504624826388638450776523592, 5.01968135180339698653294960296, 5.41340289830795980497616333727, 6.40713018566319065782761915503, 7.15583529024822094542501370686, 8.106170928651121440047566664440, 8.664044103642081964361887443345