| L(s) = 1 | − 2·3-s + 2·4-s + 9-s − 4·12-s + 4·13-s + 13·19-s − 25-s + 4·27-s − 8·31-s + 2·36-s + 4·37-s − 8·39-s + 10·43-s − 6·49-s + 8·52-s − 26·57-s − 9·61-s − 8·64-s + 15·67-s + 13·73-s + 2·75-s + 26·76-s − 20·79-s − 11·81-s + 16·93-s + 10·97-s − 2·100-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 4-s + 1/3·9-s − 1.15·12-s + 1.10·13-s + 2.98·19-s − 1/5·25-s + 0.769·27-s − 1.43·31-s + 1/3·36-s + 0.657·37-s − 1.28·39-s + 1.52·43-s − 6/7·49-s + 1.10·52-s − 3.44·57-s − 1.15·61-s − 64-s + 1.83·67-s + 1.52·73-s + 0.230·75-s + 2.98·76-s − 2.25·79-s − 1.22·81-s + 1.65·93-s + 1.01·97-s − 1/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257481 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.631495548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.631495548\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 14 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 61 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 83 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−9.183901278850318109489427234300, −8.325992317005834164299897199369, −7.87012532515414862949451391427, −7.25081623538713039423789615188, −7.14482929819650275767358812708, −6.45219310727081896814230233135, −5.94752368310629651840569376589, −5.69542369752285879895722755903, −5.19313455757641674634526330188, −4.62581839283308793970306899602, −3.75029568283948804476738749556, −3.25849041826511149887280269371, −2.60702096202400834840714577720, −1.61223534551344215906297635431, −0.879705736722903077499086320215,
0.879705736722903077499086320215, 1.61223534551344215906297635431, 2.60702096202400834840714577720, 3.25849041826511149887280269371, 3.75029568283948804476738749556, 4.62581839283308793970306899602, 5.19313455757641674634526330188, 5.69542369752285879895722755903, 5.94752368310629651840569376589, 6.45219310727081896814230233135, 7.14482929819650275767358812708, 7.25081623538713039423789615188, 7.87012532515414862949451391427, 8.325992317005834164299897199369, 9.183901278850318109489427234300
