| L(s) = 1 | + 8·2-s + 2·3-s + 34·4-s − 10·5-s + 16·6-s − 14·7-s + 96·8-s − 19·9-s − 80·10-s − 14·11-s + 68·12-s + 50·13-s − 112·14-s − 20·15-s + 196·16-s − 50·17-s − 152·18-s + 36·19-s − 340·20-s − 28·21-s − 112·22-s + 244·23-s + 192·24-s + 75·25-s + 400·26-s − 30·27-s − 476·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 0.384·3-s + 17/4·4-s − 0.894·5-s + 1.08·6-s − 0.755·7-s + 4.24·8-s − 0.703·9-s − 2.52·10-s − 0.383·11-s + 1.63·12-s + 1.06·13-s − 2.13·14-s − 0.344·15-s + 3.06·16-s − 0.713·17-s − 1.99·18-s + 0.434·19-s − 3.80·20-s − 0.290·21-s − 1.08·22-s + 2.21·23-s + 1.63·24-s + 3/5·25-s + 3.01·26-s − 0.213·27-s − 3.21·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.223089146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.223089146\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p^{3} T + 15 p T^{2} - p^{6} T^{3} + p^{6} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - 2 T + 23 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 14 T + 663 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 50 T + 4987 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 50 T + 387 p T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 36 T + 10170 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 244 T + 29970 T^{2} - 244 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 26 T + 47795 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 120 T - 1618 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 564 T + 173630 T^{2} - 564 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 p T + 133986 T^{2} + 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 260 T + 166666 T^{2} + 260 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 350 T + 203423 T^{2} + 350 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 56 T + 265770 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 616 T + p^{3} T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 336 T + 458858 T^{2} - 336 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 152 T + 599110 T^{2} + 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 952 T + p^{3} T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 676 T + 655606 T^{2} - 676 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1014 T + 1120119 T^{2} - 1014 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 376 T + 458918 T^{2} + 376 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 216 T + 1417730 T^{2} + 216 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2742 T + 3608187 T^{2} - 2742 p^{3} T^{3} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.96937458453203323925251089762, −15.36080296278189108756728642990, −14.80723333190656598792553549478, −14.78087386236095405578224077907, −13.65726969307803711827536505876, −13.46188577452627963038114562248, −12.93596040644060945302808781536, −12.55080475015299726901144508149, −11.49866612153758956587525004771, −11.43979415134699257168741039487, −10.63887138374207418976842958442, −9.307324458736904324545130481123, −8.534471157471106185264744217798, −7.52985839123901122492131590224, −6.59579649322808839307610539272, −5.99925029535903665821127577129, −5.05268532606638870053803240027, −4.37067012632211651815365282126, −3.21748054548271073994916860722, −3.10872113048333480481099347312,
3.10872113048333480481099347312, 3.21748054548271073994916860722, 4.37067012632211651815365282126, 5.05268532606638870053803240027, 5.99925029535903665821127577129, 6.59579649322808839307610539272, 7.52985839123901122492131590224, 8.534471157471106185264744217798, 9.307324458736904324545130481123, 10.63887138374207418976842958442, 11.43979415134699257168741039487, 11.49866612153758956587525004771, 12.55080475015299726901144508149, 12.93596040644060945302808781536, 13.46188577452627963038114562248, 13.65726969307803711827536505876, 14.78087386236095405578224077907, 14.80723333190656598792553549478, 15.36080296278189108756728642990, 15.96937458453203323925251089762
