L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 3·11-s − 12-s + 5·13-s + 16-s − 4·17-s − 18-s − 19-s − 3·22-s − 23-s + 24-s − 5·26-s − 27-s − 2·29-s − 31-s − 32-s − 3·33-s + 4·34-s + 36-s + 38-s − 5·39-s − 4·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.904·11-s − 0.288·12-s + 1.38·13-s + 1/4·16-s − 0.970·17-s − 0.235·18-s − 0.229·19-s − 0.639·22-s − 0.208·23-s + 0.204·24-s − 0.980·26-s − 0.192·27-s − 0.371·29-s − 0.179·31-s − 0.176·32-s − 0.522·33-s + 0.685·34-s + 1/6·36-s + 0.162·38-s − 0.800·39-s − 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.65101901153491, −12.96952465683509, −12.81017722381426, −12.00307589766788, −11.51706309515296, −11.31307062960896, −10.84703932821310, −10.27010387336198, −9.864793330071010, −9.193822734506741, −8.855697915146970, −8.391706122512250, −7.906402053477831, −7.116613227945523, −6.794982502823678, −6.214911116982794, −5.988614108097642, −5.238675425043316, −4.557072740225839, −4.014299126409135, −3.491552400360659, −2.830886695867832, −1.784854733771343, −1.626984913747928, −0.7708204395763947, 0,
0.7708204395763947, 1.626984913747928, 1.784854733771343, 2.830886695867832, 3.491552400360659, 4.014299126409135, 4.557072740225839, 5.238675425043316, 5.988614108097642, 6.214911116982794, 6.794982502823678, 7.116613227945523, 7.906402053477831, 8.391706122512250, 8.855697915146970, 9.193822734506741, 9.864793330071010, 10.27010387336198, 10.84703932821310, 11.31307062960896, 11.51706309515296, 12.00307589766788, 12.81017722381426, 12.96952465683509, 13.65101901153491