| L(s) = 1 | − 128·2-s + 952·3-s + 1.22e4·4-s + 3.20e4·5-s − 1.21e5·6-s + 2.35e5·7-s − 1.04e6·8-s − 2.10e6·9-s − 4.09e6·10-s − 1.35e6·11-s + 1.16e7·12-s + 3.51e6·13-s − 3.01e7·14-s + 3.04e7·15-s + 8.38e7·16-s + 2.17e8·17-s + 2.69e8·18-s + 5.91e8·19-s + 3.93e8·20-s + 2.24e8·21-s + 1.73e8·22-s + 8.40e8·23-s − 9.98e8·24-s − 8.35e7·25-s − 4.49e8·26-s − 3.35e9·27-s + 2.89e9·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.753·3-s + 3/2·4-s + 0.916·5-s − 1.06·6-s + 0.755·7-s − 1.41·8-s − 1.32·9-s − 1.29·10-s − 0.230·11-s + 1.13·12-s + 0.201·13-s − 1.06·14-s + 0.690·15-s + 5/4·16-s + 2.18·17-s + 1.87·18-s + 2.88·19-s + 1.37·20-s + 0.569·21-s + 0.325·22-s + 1.18·23-s − 1.06·24-s − 0.0684·25-s − 0.285·26-s − 1.66·27-s + 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(2.488071195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.488071195\) |
| \(L(\frac{15}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{6} T )^{2} \) |
| 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
−16.85919165503121449170139691220, −16.32967211634106424258565346577, −15.39665392869149713046844934397, −14.56420954578669134812072842553, −14.08468152455767563055538248503, −13.59597648215706240141784981397, −12.07090609018367119565433109264, −11.66861990635487541037459869418, −10.75944713425200007287647172559, −9.927883485928901000490812533613, −9.273817844851678909029607325188, −8.760093443543271052929621111782, −7.64182208453820525996811146247, −7.57375133227925065564901946608, −5.64920381359401383660278200266, −5.57890068998587350135203298024, −3.14378299594429020025744095199, −2.76894724655451530084241449582, −1.44925018076931525744061969641, −0.854299116562701416991740672694,
0.854299116562701416991740672694, 1.44925018076931525744061969641, 2.76894724655451530084241449582, 3.14378299594429020025744095199, 5.57890068998587350135203298024, 5.64920381359401383660278200266, 7.57375133227925065564901946608, 7.64182208453820525996811146247, 8.760093443543271052929621111782, 9.273817844851678909029607325188, 9.927883485928901000490812533613, 10.75944713425200007287647172559, 11.66861990635487541037459869418, 12.07090609018367119565433109264, 13.59597648215706240141784981397, 14.08468152455767563055538248503, 14.56420954578669134812072842553, 15.39665392869149713046844934397, 16.32967211634106424258565346577, 16.85919165503121449170139691220
