| L(s) = 1 | + 14·5-s − 22·9-s + 40·11-s − 168·19-s + 71·25-s + 12·29-s − 448·31-s + 532·41-s − 308·45-s + 610·49-s + 560·55-s − 56·59-s + 364·61-s + 816·71-s + 96·79-s − 245·81-s − 3.05e3·89-s − 2.35e3·95-s − 880·99-s + 2.49e3·101-s + 1.80e3·109-s − 1.46e3·121-s − 756·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1.25·5-s − 0.814·9-s + 1.09·11-s − 2.02·19-s + 0.567·25-s + 0.0768·29-s − 2.59·31-s + 2.02·41-s − 1.02·45-s + 1.77·49-s + 1.37·55-s − 0.123·59-s + 0.764·61-s + 1.36·71-s + 0.136·79-s − 0.336·81-s − 3.63·89-s − 2.54·95-s − 0.893·99-s + 2.45·101-s + 1.58·109-s − 1.09·121-s − 0.540·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.277048775\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.277048775\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 14 T + p^{3} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 22 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 36 T + p^{3} T^{2} )( 1 + 36 T + p^{3} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 1658 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4962 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 84 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 212 T + p^{3} T^{2} )( 1 + 212 T + p^{3} T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 224 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 86410 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 266 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 65914 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 67122 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 163690 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 28 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 182 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 419050 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 408 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 390542 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 48 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1103370 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1526 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1514050 T^{2} + p^{6} T^{4} \) |
| 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
−18.08083386241452452491186441512, −17.43398165083592770586898744393, −16.80385419446652342359555965183, −16.78745847343915461491075569813, −15.57366033980970387120580076788, −14.61699373435934343023098856324, −14.45371790187605513640779832026, −13.75028849132029244324851523436, −12.84060968202092836931152176590, −12.46815259895530776601873678406, −11.25278276316939258229292415292, −10.84802298015777363479779749574, −9.846903727540078403039718758298, −9.068424520127858991695121819413, −8.608894150981030211255287408288, −7.20124157754998590148797684238, −6.19234148341793526883379278252, −5.60960235659322356340998779670, −4.04559734726076053379783345269, −2.17574836811444390244578782448,
2.17574836811444390244578782448, 4.04559734726076053379783345269, 5.60960235659322356340998779670, 6.19234148341793526883379278252, 7.20124157754998590148797684238, 8.608894150981030211255287408288, 9.068424520127858991695121819413, 9.846903727540078403039718758298, 10.84802298015777363479779749574, 11.25278276316939258229292415292, 12.46815259895530776601873678406, 12.84060968202092836931152176590, 13.75028849132029244324851523436, 14.45371790187605513640779832026, 14.61699373435934343023098856324, 15.57366033980970387120580076788, 16.78745847343915461491075569813, 16.80385419446652342359555965183, 17.43398165083592770586898744393, 18.08083386241452452491186441512
