| L(s) = 1 | − 2·2-s + 3·4-s + 7-s − 4·8-s + 2·9-s − 11-s + 13-s − 2·14-s + 5·16-s − 4·18-s − 19-s + 2·22-s + 23-s − 2·26-s + 3·28-s − 6·32-s + 6·36-s + 37-s + 2·38-s − 41-s − 3·44-s − 2·46-s + 47-s + 3·52-s + 53-s − 4·56-s − 59-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3·4-s + 7-s − 4·8-s + 2·9-s − 11-s + 13-s − 2·14-s + 5·16-s − 4·18-s − 19-s + 2·22-s + 23-s − 2·26-s + 3·28-s − 6·32-s + 6·36-s + 37-s + 2·38-s − 41-s − 3·44-s − 2·46-s + 47-s + 3·52-s + 53-s − 4·56-s − 59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5599360151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5599360151\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 7 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 13 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 23 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 41 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 53 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 59 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−10.21909375937087143863763296476, −10.20578919496777106503512875376, −9.336281646077878582184375078988, −9.320152605592396072850876376505, −8.606313541949706043374262076575, −8.428246276236042137224037194028, −7.85747715956589020015326724728, −7.68557501242738725718451301627, −7.03934487728620344713907360437, −7.03752568906865511043859680013, −6.17645518178443741231121633156, −6.12132479717912161917720685954, −5.15686831524762743437841247795, −4.88204092378953947586724392107, −4.03052308413741313298095471719, −3.62933963483017492191714838091, −2.67523889781754086218996491939, −2.26435807853597909906209284417, −1.45685872744034564526758530466, −1.18834119397106713934378917459,
1.18834119397106713934378917459, 1.45685872744034564526758530466, 2.26435807853597909906209284417, 2.67523889781754086218996491939, 3.62933963483017492191714838091, 4.03052308413741313298095471719, 4.88204092378953947586724392107, 5.15686831524762743437841247795, 6.12132479717912161917720685954, 6.17645518178443741231121633156, 7.03752568906865511043859680013, 7.03934487728620344713907360437, 7.68557501242738725718451301627, 7.85747715956589020015326724728, 8.428246276236042137224037194028, 8.606313541949706043374262076575, 9.320152605592396072850876376505, 9.336281646077878582184375078988, 10.20578919496777106503512875376, 10.21909375937087143863763296476
