| L(s) = 1 | − 952·3-s + 3.20e4·5-s − 2.35e5·7-s − 2.10e6·9-s + 1.35e6·11-s + 3.51e6·13-s − 3.04e7·15-s + 2.17e8·17-s − 5.91e8·19-s + 2.24e8·21-s − 8.40e8·23-s − 8.35e7·25-s + 3.35e9·27-s − 4.87e8·29-s − 2.19e9·31-s − 1.28e9·33-s − 7.53e9·35-s + 4.05e8·37-s − 3.34e9·39-s + 8.51e9·41-s − 2.62e10·43-s − 6.74e10·45-s − 1.55e11·47-s + 4.15e10·49-s − 2.07e11·51-s + 6.60e10·53-s + 4.32e10·55-s + ⋯ |
| L(s) = 1 | − 0.753·3-s + 0.916·5-s − 0.755·7-s − 1.32·9-s + 0.230·11-s + 0.201·13-s − 0.690·15-s + 2.18·17-s − 2.88·19-s + 0.569·21-s − 1.18·23-s − 0.0684·25-s + 1.66·27-s − 0.152·29-s − 0.443·31-s − 0.173·33-s − 0.692·35-s + 0.0259·37-s − 0.152·39-s + 0.280·41-s − 0.632·43-s − 1.21·45-s − 2.09·47-s + 3/7·49-s − 1.64·51-s + 0.409·53-s + 0.210·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.995495810\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.995495810\) |
| \(L(\frac{15}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p^{6} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 952 T + 1004902 p T^{2} + 952 p^{13} T^{3} + p^{26} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 32004 T + 8862578 p^{3} T^{2} - 32004 p^{13} T^{3} + p^{26} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 122976 p T + 167454226342 p^{2} T^{2} - 122976 p^{14} T^{3} + p^{26} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3510388 T - 428626086166758 T^{2} - 3510388 p^{13} T^{3} + p^{26} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 217711956 T + 25067368928085622 T^{2} - 217711956 p^{13} T^{3} + p^{26} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 591335752 T + 166308567245286738 T^{2} + 591335752 p^{13} T^{3} + p^{26} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 840735000 T + 700729767591174766 T^{2} + 840735000 p^{13} T^{3} + p^{26} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 487623540 T + 18294663649280992942 T^{2} + 487623540 p^{13} T^{3} + p^{26} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2193076144 T + 45113561711005815150 T^{2} + 2193076144 p^{13} T^{3} + p^{26} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 405060268 T + \)\(44\!\cdots\!74\)\( T^{2} - 405060268 p^{13} T^{3} + p^{26} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 207760308 p T + \)\(79\!\cdots\!34\)\( T^{2} - 207760308 p^{14} T^{3} + p^{26} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 26225045296 T + \)\(30\!\cdots\!54\)\( T^{2} + 26225045296 p^{13} T^{3} + p^{26} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 155048849760 T + \)\(16\!\cdots\!30\)\( T^{2} + 155048849760 p^{13} T^{3} + p^{26} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 66007050492 T + \)\(69\!\cdots\!38\)\( T^{2} - 66007050492 p^{13} T^{3} + p^{26} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 476362296984 T + \)\(24\!\cdots\!26\)\( T^{2} - 476362296984 p^{13} T^{3} + p^{26} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 197378850004 T + \)\(26\!\cdots\!50\)\( T^{2} - 197378850004 p^{13} T^{3} + p^{26} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 1718732859488 T + \)\(15\!\cdots\!66\)\( T^{2} - 1718732859488 p^{13} T^{3} + p^{26} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 695543478336 T + \)\(11\!\cdots\!62\)\( T^{2} - 695543478336 p^{13} T^{3} + p^{26} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 466085239340 T + \)\(25\!\cdots\!82\)\( T^{2} + 466085239340 p^{13} T^{3} + p^{26} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2432016575840 T + \)\(82\!\cdots\!74\)\( T^{2} - 2432016575840 p^{13} T^{3} + p^{26} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1743984494616 T + \)\(17\!\cdots\!94\)\( T^{2} - 1743984494616 p^{13} T^{3} + p^{26} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3022580240484 T + \)\(43\!\cdots\!18\)\( T^{2} - 3022580240484 p^{13} T^{3} + p^{26} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 7760062661092 T + \)\(14\!\cdots\!14\)\( T^{2} - 7760062661092 p^{13} T^{3} + p^{26} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.34378319390049801479128676999, −10.88649367442267786598332155027, −10.10610694959025056805545774634, −10.08529910243158558859014012154, −9.413275007856369137611130143226, −8.638068977803631725364766097791, −8.323526076917495904587002159890, −7.78250564130892244832508217356, −6.57779478823098993293237328358, −6.50109565568670337786593592358, −5.94462863336374883663503054530, −5.51518740480149734623907355300, −5.07645185474176128424820594101, −4.06267005263107272787778060199, −3.53644644342061064698586881363, −2.93674312885581899436411509601, −2.00742159693441648058117147089, −1.93149658748131401884387344423, −0.57289381993979388905357424082, −0.52564426835011462282423029224,
0.52564426835011462282423029224, 0.57289381993979388905357424082, 1.93149658748131401884387344423, 2.00742159693441648058117147089, 2.93674312885581899436411509601, 3.53644644342061064698586881363, 4.06267005263107272787778060199, 5.07645185474176128424820594101, 5.51518740480149734623907355300, 5.94462863336374883663503054530, 6.50109565568670337786593592358, 6.57779478823098993293237328358, 7.78250564130892244832508217356, 8.323526076917495904587002159890, 8.638068977803631725364766097791, 9.413275007856369137611130143226, 10.08529910243158558859014012154, 10.10610694959025056805545774634, 10.88649367442267786598332155027, 11.34378319390049801479128676999
