| L(s) = 1 | − 2·4-s + 2·7-s − 34·13-s + 4·16-s − 62·19-s − 4·28-s − 62·31-s + 32·37-s − 130·43-s − 95·49-s + 68·52-s − 110·61-s − 8·64-s + 146·67-s − 112·73-s + 124·76-s + 208·79-s − 68·91-s − 178·97-s + 164·103-s + 142·109-s + 8·112-s + 224·121-s + 124·124-s + 127-s + 131-s − 124·133-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2/7·7-s − 2.61·13-s + 1/4·16-s − 3.26·19-s − 1/7·28-s − 2·31-s + 0.864·37-s − 3.02·43-s − 1.93·49-s + 1.30·52-s − 1.80·61-s − 1/8·64-s + 2.17·67-s − 1.53·73-s + 1.63·76-s + 2.63·79-s − 0.747·91-s − 1.83·97-s + 1.59·103-s + 1.30·109-s + 1/14·112-s + 1.85·121-s + 124-s + 0.00787·127-s + 0.00763·131-s − 0.932·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.04328956722\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04328956722\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 224 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 17 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 304 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 31 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1040 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 224 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3074 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 65 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3968 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 5330 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 6160 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 55 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 73 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 4030 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 56 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 104 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 13616 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 14690 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 89 T + p^{2} T^{2} )^{2} \) |
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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
−11.33642974475651924715414587639, −10.40128465680970104126951091086, −10.36722855205109480451567753319, −9.559875431069365339770261971002, −9.558666920018565810649691703204, −8.772597990345966157704522536671, −8.446858878567979726205913223730, −7.87987907715483471726161293459, −7.59155722306616180575732613003, −6.72235265815396640297665690620, −6.69416972794422562389796132019, −5.92435429695627644534200802953, −5.17443364505919745754106465289, −4.77510706888705989713177455992, −4.51421032038097734486621187700, −3.76908498611875277689823642218, −3.04131887265951134183950992105, −2.05644225840217699354100113888, −1.95745113347153161238102852244, −0.080853202810071802739900002130,
0.080853202810071802739900002130, 1.95745113347153161238102852244, 2.05644225840217699354100113888, 3.04131887265951134183950992105, 3.76908498611875277689823642218, 4.51421032038097734486621187700, 4.77510706888705989713177455992, 5.17443364505919745754106465289, 5.92435429695627644534200802953, 6.69416972794422562389796132019, 6.72235265815396640297665690620, 7.59155722306616180575732613003, 7.87987907715483471726161293459, 8.446858878567979726205913223730, 8.772597990345966157704522536671, 9.558666920018565810649691703204, 9.559875431069365339770261971002, 10.36722855205109480451567753319, 10.40128465680970104126951091086, 11.33642974475651924715414587639
