| L(s) = 1 | − 4·4-s + 6·5-s + 12·16-s − 24·20-s + 18·23-s + 17·25-s − 10·31-s + 14·37-s + 24·47-s − 14·49-s − 12·53-s + 30·59-s − 32·64-s + 26·67-s + 6·71-s + 72·80-s + 18·89-s − 72·92-s + 34·97-s − 68·100-s − 8·103-s − 42·113-s + 108·115-s + 40·124-s + 18·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 2·4-s + 2.68·5-s + 3·16-s − 5.36·20-s + 3.75·23-s + 17/5·25-s − 1.79·31-s + 2.30·37-s + 3.50·47-s − 2·49-s − 1.64·53-s + 3.90·59-s − 4·64-s + 3.17·67-s + 0.712·71-s + 8.04·80-s + 1.90·89-s − 7.50·92-s + 3.45·97-s − 6.79·100-s − 0.788·103-s − 3.95·113-s + 10.0·115-s + 3.59·124-s + 1.60·125-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.795545521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.795545521\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
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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
−12.52465149227571197001763678656, −11.27422597650334929128966428619, −11.27422597650334929128966428619, −10.49934084603472471085153211433, −10.49934084603472471085153211433, −9.638742540829787389319183524300, −9.638742540829787389319183524300, −9.239645513051175281393608873936, −9.239645513051175281393608873936, −8.468596719955261371677845250157, −8.468596719955261371677845250157, −7.37520405553462112512317968814, −7.37520405553462112512317968814, −6.34583551331242224592183459087, −6.34583551331242224592183459087, −5.46569305050405475214160980267, −5.46569305050405475214160980267, −4.87628538913627800967773832068, −4.87628538913627800967773832068, −3.66923064750984294129055958828, −3.66923064750984294129055958828, −2.44495639562721598403091577465, −2.44495639562721598403091577465, −1.06294681607902093882729364169, −1.06294681607902093882729364169,
1.06294681607902093882729364169, 1.06294681607902093882729364169, 2.44495639562721598403091577465, 2.44495639562721598403091577465, 3.66923064750984294129055958828, 3.66923064750984294129055958828, 4.87628538913627800967773832068, 4.87628538913627800967773832068, 5.46569305050405475214160980267, 5.46569305050405475214160980267, 6.34583551331242224592183459087, 6.34583551331242224592183459087, 7.37520405553462112512317968814, 7.37520405553462112512317968814, 8.468596719955261371677845250157, 8.468596719955261371677845250157, 9.239645513051175281393608873936, 9.239645513051175281393608873936, 9.638742540829787389319183524300, 9.638742540829787389319183524300, 10.49934084603472471085153211433, 10.49934084603472471085153211433, 11.27422597650334929128966428619, 11.27422597650334929128966428619, 12.52465149227571197001763678656
