| L(s) = 1 | − 3·4-s + 18·7-s − 20·13-s − 7·16-s − 58·19-s − 54·28-s + 30·31-s − 118·37-s + 10·43-s + 145·49-s + 60·52-s + 38·61-s + 69·64-s + 108·67-s − 110·73-s + 174·76-s − 66·79-s − 360·91-s + 194·97-s + 50·103-s − 122·109-s − 126·112-s + 66·121-s − 90·124-s + 127-s + 131-s − 1.04e3·133-s + ⋯ |
| L(s) = 1 | − 3/4·4-s + 18/7·7-s − 1.53·13-s − 0.437·16-s − 3.05·19-s − 1.92·28-s + 0.967·31-s − 3.18·37-s + 0.232·43-s + 2.95·49-s + 1.15·52-s + 0.622·61-s + 1.07·64-s + 1.61·67-s − 1.50·73-s + 2.28·76-s − 0.835·79-s − 3.95·91-s + 2·97-s + 0.485·103-s − 1.11·109-s − 9/8·112-s + 6/11·121-s − 0.725·124-s + 0.00787·127-s + 0.00763·131-s − 7.84·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.215998605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.215998605\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 9 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 p T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 126 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 29 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 526 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 978 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 15 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 59 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3186 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4242 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 5442 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 2562 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 19 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 54 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 1458 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 55 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 33 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 7442 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 1586 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| 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
−10.50536488320776168496330695529, −10.14823351107184938612534121395, −9.769819900067945020826146039736, −8.794227216109497953791708902057, −8.791441489294073314315535266062, −8.497742084295406044049560595158, −7.994216707629330530745734697772, −7.62967650716133835526829971150, −6.87449097391103656838588883633, −6.79277259916902643950169359660, −5.91199562877668563926999302500, −5.25453309538724487439144073430, −4.94210219285211579239811017237, −4.49528023258761387059686232560, −4.38196453132181319019603141050, −3.62375907577974790955259686730, −2.48912712902769828751083152618, −2.04089887554383088700392320320, −1.65303521512403699077205890954, −0.38770688746401793440784690704,
0.38770688746401793440784690704, 1.65303521512403699077205890954, 2.04089887554383088700392320320, 2.48912712902769828751083152618, 3.62375907577974790955259686730, 4.38196453132181319019603141050, 4.49528023258761387059686232560, 4.94210219285211579239811017237, 5.25453309538724487439144073430, 5.91199562877668563926999302500, 6.79277259916902643950169359660, 6.87449097391103656838588883633, 7.62967650716133835526829971150, 7.994216707629330530745734697772, 8.497742084295406044049560595158, 8.791441489294073314315535266062, 8.794227216109497953791708902057, 9.769819900067945020826146039736, 10.14823351107184938612534121395, 10.50536488320776168496330695529
