| L(s) = 1 | + 2·3-s + 2·7-s + 6·11-s + 2·13-s + 6·17-s + 4·19-s + 4·21-s + 6·23-s − 7·25-s − 2·27-s − 6·29-s − 8·31-s + 12·33-s − 2·37-s + 4·39-s − 14·43-s + 12·47-s + 3·49-s + 12·51-s − 6·53-s + 8·57-s − 20·61-s − 8·67-s + 12·69-s − 6·71-s − 8·73-s − 14·75-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.755·7-s + 1.80·11-s + 0.554·13-s + 1.45·17-s + 0.917·19-s + 0.872·21-s + 1.25·23-s − 7/5·25-s − 0.384·27-s − 1.11·29-s − 1.43·31-s + 2.08·33-s − 0.328·37-s + 0.640·39-s − 2.13·43-s + 1.75·47-s + 3/7·49-s + 1.68·51-s − 0.824·53-s + 1.05·57-s − 2.56·61-s − 0.977·67-s + 1.44·69-s − 0.712·71-s − 0.936·73-s − 1.61·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 132496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.929934528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.929934528\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 16 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 15 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 55 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 51 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 48 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 127 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 67 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 135 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 51 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 175 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 187 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 231 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−11.55354777303285183537828862048, −11.48162942371729126931923445671, −10.62562328018689743867872336853, −10.36084466657607284142720822721, −9.427953620658775691068582805548, −9.343634114620225412128553460420, −8.919002538328781227320795980179, −8.561716837758046088552720669195, −7.80985016415399555533128753352, −7.60300856211437116905583953285, −7.14403758966548007283223252676, −6.39449945158992490277324879979, −5.73070678677033449952453341354, −5.44610844428662567205651774747, −4.58578199167383713369794242349, −3.90423520744475782645598411607, −3.35473243646736378058596595982, −3.08037979132238109847459385699, −1.76859295600357545086886342204, −1.40394088187422053685394686066,
1.40394088187422053685394686066, 1.76859295600357545086886342204, 3.08037979132238109847459385699, 3.35473243646736378058596595982, 3.90423520744475782645598411607, 4.58578199167383713369794242349, 5.44610844428662567205651774747, 5.73070678677033449952453341354, 6.39449945158992490277324879979, 7.14403758966548007283223252676, 7.60300856211437116905583953285, 7.80985016415399555533128753352, 8.561716837758046088552720669195, 8.919002538328781227320795980179, 9.343634114620225412128553460420, 9.427953620658775691068582805548, 10.36084466657607284142720822721, 10.62562328018689743867872336853, 11.48162942371729126931923445671, 11.55354777303285183537828862048
