L(s) = 1 | − 2·3-s + 6·7-s + 5·9-s − 2·11-s − 4·13-s + 6·17-s + 6·19-s − 12·21-s − 6·23-s − 4·25-s − 10·27-s − 6·29-s + 4·33-s − 2·37-s + 8·39-s + 6·41-s − 6·43-s − 24·47-s + 21·49-s − 12·51-s − 12·57-s + 6·59-s − 14·61-s + 30·63-s + 18·67-s + 12·69-s − 6·71-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 2.26·7-s + 5/3·9-s − 0.603·11-s − 1.10·13-s + 1.45·17-s + 1.37·19-s − 2.61·21-s − 1.25·23-s − 4/5·25-s − 1.92·27-s − 1.11·29-s + 0.696·33-s − 0.328·37-s + 1.28·39-s + 0.937·41-s − 0.914·43-s − 3.50·47-s + 3·49-s − 1.68·51-s − 1.58·57-s + 0.781·59-s − 1.79·61-s + 3.77·63-s + 2.19·67-s + 1.44·69-s − 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.836494168\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.836494168\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
good | 3 | $D_4\times C_2$ | \( 1 + 2 T - T^{2} - 2 T^{3} + 4 T^{4} - 2 p T^{5} - p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 6 T + 15 T^{2} - 6 p T^{3} + 20 p T^{4} - 6 p^{2} T^{5} + 15 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 2 T - T^{2} - 34 T^{3} - 140 T^{4} - 34 p T^{5} - p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 6 T + T^{2} - 6 T^{3} + 324 T^{4} - 6 p T^{5} + p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 6 T + 7 T^{2} + 54 T^{3} - 204 T^{4} + 54 p T^{5} + 7 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T - T^{2} - 54 T^{3} + 12 T^{4} - 54 p T^{5} - p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 6 T + T^{2} - 138 T^{3} - 660 T^{4} - 138 p T^{5} + p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 2 T + T^{2} - 142 T^{3} - 1508 T^{4} - 142 p T^{5} + p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 6 T - 47 T^{2} - 6 T^{3} + 3732 T^{4} - 6 p T^{5} - 47 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 6 T - 9 T^{2} - 246 T^{3} - 1372 T^{4} - 246 p T^{5} - 9 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 6 T - 73 T^{2} + 54 T^{3} + 6276 T^{4} + 54 p T^{5} - 73 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 18 T + 127 T^{2} - 1134 T^{3} + 12612 T^{4} - 1134 p T^{5} + 127 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 6 T - 97 T^{2} - 54 T^{3} + 10092 T^{4} - 54 p T^{5} - 97 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 89 | $D_4\times C_2$ | \( 1 - 18 T + 97 T^{2} - 882 T^{3} + 14772 T^{4} - 882 p T^{5} + 97 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 9 T - 16 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.020247026217916835361788042620, −7.890454310843357315933530154568, −7.76604895823585456810387293220, −7.32545419090055812618387222217, −7.17435001523400540384268839752, −7.16301982154919587188416679669, −6.44681822157245728736762298299, −6.43386664599844950863617194973, −6.10644366400052375453825882071, −5.60551626322680876268635154308, −5.48198528747405048092641515389, −5.35031050011928999273122749769, −5.03758832816359361039200034102, −4.85611813310988074459383364679, −4.66746644552499408139395255332, −4.25630128671750165758581541419, −4.02518619012058607441005479869, −3.46648177626219165005588939596, −3.39697408338966575749145793737, −2.97781434624446805719630255174, −2.04460082587664742649847813553, −1.92858244175304535860187854915, −1.92262402817580827586058454024, −1.18902363818117412411488506714, −0.59675671118218918460236937320,
0.59675671118218918460236937320, 1.18902363818117412411488506714, 1.92262402817580827586058454024, 1.92858244175304535860187854915, 2.04460082587664742649847813553, 2.97781434624446805719630255174, 3.39697408338966575749145793737, 3.46648177626219165005588939596, 4.02518619012058607441005479869, 4.25630128671750165758581541419, 4.66746644552499408139395255332, 4.85611813310988074459383364679, 5.03758832816359361039200034102, 5.35031050011928999273122749769, 5.48198528747405048092641515389, 5.60551626322680876268635154308, 6.10644366400052375453825882071, 6.43386664599844950863617194973, 6.44681822157245728736762298299, 7.16301982154919587188416679669, 7.17435001523400540384268839752, 7.32545419090055812618387222217, 7.76604895823585456810387293220, 7.890454310843357315933530154568, 8.020247026217916835361788042620