| L(s) = 1 | − 2·3-s + 3·9-s + 12·11-s + 12·17-s + 10·19-s − 4·27-s − 24·33-s + 2·43-s − 13·49-s − 24·51-s − 20·57-s − 12·59-s − 22·67-s − 4·73-s + 5·81-s + 12·83-s + 14·97-s + 36·99-s − 24·107-s − 24·113-s + 86·121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s + 26·147-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s + 3.61·11-s + 2.91·17-s + 2.29·19-s − 0.769·27-s − 4.17·33-s + 0.304·43-s − 1.85·49-s − 3.36·51-s − 2.64·57-s − 1.56·59-s − 2.68·67-s − 0.468·73-s + 5/9·81-s + 1.31·83-s + 1.42·97-s + 3.61·99-s − 2.32·107-s − 2.25·113-s + 7.81·121-s + 0.0887·127-s − 0.352·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.14·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.546582227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.546582227\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
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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
−7.65692053420996459201778762115, −7.54038315127717275337161988087, −6.94607178869965744891714265309, −6.58837472702591983831965532194, −6.16571856132998816140053150198, −5.67959326371779885195862676134, −5.57958254615529070693398763832, −4.74416118025497742788913645945, −4.48317997032114878344902590623, −3.73650586061389001500010789061, −3.37887843303378899402222918182, −3.14969814812418521417534836385, −1.52862254168278113235246302632, −1.35841755750539912162882790880, −0.942550376347668032286962617546,
0.942550376347668032286962617546, 1.35841755750539912162882790880, 1.52862254168278113235246302632, 3.14969814812418521417534836385, 3.37887843303378899402222918182, 3.73650586061389001500010789061, 4.48317997032114878344902590623, 4.74416118025497742788913645945, 5.57958254615529070693398763832, 5.67959326371779885195862676134, 6.16571856132998816140053150198, 6.58837472702591983831965532194, 6.94607178869965744891714265309, 7.54038315127717275337161988087, 7.65692053420996459201778762115
