L(s) = 1 | − 5-s + 4·7-s − 6·13-s − 6·17-s + 4·19-s + 4·23-s + 25-s − 2·29-s − 8·31-s − 4·35-s − 10·37-s + 10·41-s + 4·47-s + 9·49-s + 10·53-s − 4·59-s + 2·61-s + 6·65-s + 8·67-s + 14·73-s − 16·79-s + 8·83-s + 6·85-s + 6·89-s − 24·91-s − 4·95-s + 2·97-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.51·7-s − 1.66·13-s − 1.45·17-s + 0.917·19-s + 0.834·23-s + 1/5·25-s − 0.371·29-s − 1.43·31-s − 0.676·35-s − 1.64·37-s + 1.56·41-s + 0.583·47-s + 9/7·49-s + 1.37·53-s − 0.520·59-s + 0.256·61-s + 0.744·65-s + 0.977·67-s + 1.63·73-s − 1.80·79-s + 0.878·83-s + 0.650·85-s + 0.635·89-s − 2.51·91-s − 0.410·95-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 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
−14.26353958079097, −13.75201804076449, −13.16861579554792, −12.51553954984180, −12.19847179732289, −11.59871374326656, −11.21661194363550, −10.78579663406048, −10.36886319338247, −9.413739320486630, −9.221400140687911, −8.632716255782946, −7.989245745590448, −7.592770213602847, −6.981936648473513, −6.897756234485526, −5.631346158362721, −5.280945833348139, −4.867540026592487, −4.267361218294928, −3.784170560696384, −2.850464185656171, −2.263019297968219, −1.769932433114861, −0.8690978881654730, 0,
0.8690978881654730, 1.769932433114861, 2.263019297968219, 2.850464185656171, 3.784170560696384, 4.267361218294928, 4.867540026592487, 5.280945833348139, 5.631346158362721, 6.897756234485526, 6.981936648473513, 7.592770213602847, 7.989245745590448, 8.632716255782946, 9.221400140687911, 9.413739320486630, 10.36886319338247, 10.78579663406048, 11.21661194363550, 11.59871374326656, 12.19847179732289, 12.51553954984180, 13.16861579554792, 13.75201804076449, 14.26353958079097