| L(s) = 1 | − 5·2-s + 16·4-s + 3·5-s − 91·7-s − 115·8-s − 15·10-s − 149·11-s + 455·14-s + 575·16-s + 462·17-s − 618·19-s + 48·20-s + 745·22-s − 560·23-s − 619·25-s − 1.45e3·28-s − 470·29-s − 2.22e3·31-s − 1.84e3·32-s − 2.31e3·34-s − 273·35-s − 1.97e3·37-s + 3.09e3·38-s − 345·40-s + 5.59e3·43-s − 2.38e3·44-s + 2.80e3·46-s + ⋯ |
| L(s) = 1 | − 5/4·2-s + 4-s + 3/25·5-s − 1.85·7-s − 1.79·8-s − 0.149·10-s − 1.23·11-s + 2.32·14-s + 2.24·16-s + 1.59·17-s − 1.71·19-s + 3/25·20-s + 1.53·22-s − 1.05·23-s − 0.990·25-s − 1.85·28-s − 0.558·29-s − 2.31·31-s − 1.79·32-s − 1.99·34-s − 0.222·35-s − 1.43·37-s + 2.13·38-s − 0.215·40-s + 3.02·43-s − 1.23·44-s + 1.32·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.004247087569\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.004247087569\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 13 p T + p^{4} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 5 T + 9 T^{2} + 5 p^{4} T^{3} + p^{8} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 628 T^{2} - 3 p^{4} T^{3} + p^{8} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 149 T + 7560 T^{2} + 149 p^{4} T^{3} + p^{8} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 55394 T^{2} + p^{8} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 462 T + 154669 T^{2} - 462 p^{4} T^{3} + p^{8} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 618 T + 257629 T^{2} + 618 p^{4} T^{3} + p^{8} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 560 T + 33759 T^{2} + 560 p^{4} T^{3} + p^{8} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 235 T + p^{4} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2229 T + 2579668 T^{2} + 2229 p^{4} T^{3} + p^{8} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 1970 T + 2006739 T^{2} + 1970 p^{4} T^{3} + p^{8} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 2280106 T^{2} + p^{8} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2798 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 7152 T + 21930049 T^{2} + 7152 p^{4} T^{3} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 17 p T - 2520 p^{2} T^{2} - 17 p^{5} T^{3} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 2391 T + 14022988 T^{2} + 2391 p^{4} T^{3} + p^{8} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6180 T + 26576641 T^{2} + 6180 p^{4} T^{3} + p^{8} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4156 T - 2878785 T^{2} - 4156 p^{4} T^{3} + p^{8} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 484 T + p^{4} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 1644 T + 29299153 T^{2} + 1644 p^{4} T^{3} + p^{8} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 4325 T - 20244456 T^{2} + 4325 p^{4} T^{3} + p^{8} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 93215615 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 3426 T + 66654733 T^{2} + 3426 p^{4} T^{3} + p^{8} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 147347735 T^{2} + p^{8} T^{4} \) |
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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
−15.38637091782331011355869843714, −14.12155271028935054030931576708, −12.90638396019745953701541575178, −12.81261087880363918595898393148, −12.42824168251302652717525502608, −11.63721444747428639343708153515, −10.79994623776290883396697862662, −10.11989537294148929026637536394, −10.03620408338163057700380584657, −9.164542980800684335566468404032, −8.958429032992665083859125947636, −7.906593050447395876025948245395, −7.57286100448278795688032252440, −6.58366858677632123181109334138, −5.91745657096087239336086127305, −5.60201185076193046955312168395, −3.72358071373524853974406315551, −3.12590000359200993694223225491, −2.06880164958918333078692348624, −0.04104565321404852694461227154,
0.04104565321404852694461227154, 2.06880164958918333078692348624, 3.12590000359200993694223225491, 3.72358071373524853974406315551, 5.60201185076193046955312168395, 5.91745657096087239336086127305, 6.58366858677632123181109334138, 7.57286100448278795688032252440, 7.906593050447395876025948245395, 8.958429032992665083859125947636, 9.164542980800684335566468404032, 10.03620408338163057700380584657, 10.11989537294148929026637536394, 10.79994623776290883396697862662, 11.63721444747428639343708153515, 12.42824168251302652717525502608, 12.81261087880363918595898393148, 12.90638396019745953701541575178, 14.12155271028935054030931576708, 15.38637091782331011355869843714
