| L(s) = 1 | − 4·3-s + 8·9-s − 16·11-s + 12·13-s − 12·27-s + 64·33-s + 20·37-s − 48·39-s + 48·47-s − 24·49-s + 8·59-s + 36·61-s + 23·81-s − 32·97-s − 128·99-s + 8·107-s − 12·109-s − 80·111-s + 96·117-s + 128·121-s + 127-s + 131-s + 137-s + 139-s − 192·141-s − 192·143-s + 96·147-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 8/3·9-s − 4.82·11-s + 3.32·13-s − 2.30·27-s + 11.1·33-s + 3.28·37-s − 7.68·39-s + 7.00·47-s − 3.42·49-s + 1.04·59-s + 4.60·61-s + 23/9·81-s − 3.24·97-s − 12.8·99-s + 0.773·107-s − 1.14·109-s − 7.59·111-s + 8.87·117-s + 11.6·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 16.1·141-s − 16.0·143-s + 7.91·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9364141368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9364141368\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )( 1 + 8 T^{2} + p^{2} T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^3$ | \( 1 - 238 T^{4} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 29 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )( 1 + 40 T^{2} + p^{2} T^{4} ) \) |
| 31 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 1202 T^{4} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 53 | $C_2^3$ | \( 1 - 718 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 7822 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 126 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 60 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 146 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.56195732810306319542609508117, −6.97643630889629602680959269716, −6.87580423069156595265883726222, −6.84026538889479409669445572057, −6.20876796155066319042683869687, −6.19703071818623312072098279957, −5.81442478922943558226500719233, −5.65003179806331772368854273771, −5.63651259836370724882022192173, −5.50304266547079785813016490708, −5.25693459462596953092024969211, −4.87644440087465223573897671446, −4.74075138846629789386363083092, −4.28020592639908912736267173512, −4.00615719471817915888535451998, −3.93393474773124816158575668856, −3.57559922872082426013509493896, −3.01578629692544678035951079452, −2.67792065654966689250641319953, −2.56205934450786844205828984305, −2.31805544006625968321969241311, −1.78871108050730256078595318024, −0.975907692775838035187846374937, −0.798361123437420315940350322513, −0.48247196254422493429404842821,
0.48247196254422493429404842821, 0.798361123437420315940350322513, 0.975907692775838035187846374937, 1.78871108050730256078595318024, 2.31805544006625968321969241311, 2.56205934450786844205828984305, 2.67792065654966689250641319953, 3.01578629692544678035951079452, 3.57559922872082426013509493896, 3.93393474773124816158575668856, 4.00615719471817915888535451998, 4.28020592639908912736267173512, 4.74075138846629789386363083092, 4.87644440087465223573897671446, 5.25693459462596953092024969211, 5.50304266547079785813016490708, 5.63651259836370724882022192173, 5.65003179806331772368854273771, 5.81442478922943558226500719233, 6.19703071818623312072098279957, 6.20876796155066319042683869687, 6.84026538889479409669445572057, 6.87580423069156595265883726222, 6.97643630889629602680959269716, 7.56195732810306319542609508117
