L(s) = 1 | − 5·7-s − 11-s − 13-s + 3·17-s − 6·19-s + 3·23-s − 4·29-s + 5·37-s − 11·41-s − 6·43-s + 18·49-s − 9·53-s − 12·59-s + 5·61-s − 8·67-s − 13·71-s + 10·73-s + 5·77-s − 17·79-s − 4·83-s + 11·89-s + 5·91-s + 7·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
L(s) = 1 | − 1.88·7-s − 0.301·11-s − 0.277·13-s + 0.727·17-s − 1.37·19-s + 0.625·23-s − 0.742·29-s + 0.821·37-s − 1.71·41-s − 0.914·43-s + 18/7·49-s − 1.23·53-s − 1.56·59-s + 0.640·61-s − 0.977·67-s − 1.54·71-s + 1.17·73-s + 0.569·77-s − 1.91·79-s − 0.439·83-s + 1.16·89-s + 0.524·91-s + 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−14.42555036805912, −13.55796373043344, −13.23414514544581, −12.98007838852271, −12.35900367285406, −12.08589137758855, −11.35568804819086, −10.73685373443339, −10.28819684137834, −9.869825522120447, −9.438882275083645, −8.891762728302543, −8.418292913658316, −7.673822651764522, −7.239335055134987, −6.533515259651553, −6.346045765403125, −5.712094998013418, −5.105504520588685, −4.419506732238021, −3.788138859683576, −3.151136801651212, −2.883403697426476, −2.050145480519638, −1.244120077933858, 0, 0,
1.244120077933858, 2.050145480519638, 2.883403697426476, 3.151136801651212, 3.788138859683576, 4.419506732238021, 5.105504520588685, 5.712094998013418, 6.346045765403125, 6.533515259651553, 7.239335055134987, 7.673822651764522, 8.418292913658316, 8.891762728302543, 9.438882275083645, 9.869825522120447, 10.28819684137834, 10.73685373443339, 11.35568804819086, 12.08589137758855, 12.35900367285406, 12.98007838852271, 13.23414514544581, 13.55796373043344, 14.42555036805912