| L(s) = 1 | − 5-s + 4·7-s − 2·13-s + 17-s + 4·19-s + 8·23-s + 25-s − 6·29-s − 4·35-s + 6·37-s − 2·41-s + 4·43-s + 8·47-s + 9·49-s + 10·53-s − 8·59-s − 6·61-s + 2·65-s + 4·67-s − 8·71-s + 10·73-s − 12·83-s − 85-s + 14·89-s − 8·91-s − 4·95-s + 10·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 0.554·13-s + 0.242·17-s + 0.917·19-s + 1.66·23-s + 1/5·25-s − 1.11·29-s − 0.676·35-s + 0.986·37-s − 0.312·41-s + 0.609·43-s + 1.16·47-s + 9/7·49-s + 1.37·53-s − 1.04·59-s − 0.768·61-s + 0.248·65-s + 0.488·67-s − 0.949·71-s + 1.17·73-s − 1.31·83-s − 0.108·85-s + 1.48·89-s − 0.838·91-s − 0.410·95-s + 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.119839201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.119839201\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−14.58763145236195, −14.22804847121788, −13.45663237360771, −13.13778307203702, −12.29537718161076, −11.97455701480988, −11.44082110532659, −10.94716534322669, −10.66026640496412, −9.831804777622605, −9.189553242550057, −8.859430274157427, −8.120207699887903, −7.516532984395673, −7.441765732700209, −6.695751190364238, −5.690109193678327, −5.375699335872891, −4.715292462782462, −4.312242876965778, −3.506360060792420, −2.819809574670146, −2.111302377080464, −1.292869300301346, −0.6869840788092505,
0.6869840788092505, 1.292869300301346, 2.111302377080464, 2.819809574670146, 3.506360060792420, 4.312242876965778, 4.715292462782462, 5.375699335872891, 5.690109193678327, 6.695751190364238, 7.441765732700209, 7.516532984395673, 8.120207699887903, 8.859430274157427, 9.189553242550057, 9.831804777622605, 10.66026640496412, 10.94716534322669, 11.44082110532659, 11.97455701480988, 12.29537718161076, 13.13778307203702, 13.45663237360771, 14.22804847121788, 14.58763145236195
