| L(s) = 1 | + 2·3-s − 2·7-s + 2·9-s − 4·11-s − 6·13-s + 2·19-s − 4·21-s − 8·23-s + 6·27-s + 4·31-s − 8·33-s − 12·39-s − 8·41-s − 4·43-s − 12·47-s + 3·49-s + 20·53-s + 4·57-s − 14·59-s + 18·61-s − 4·63-s − 8·67-s − 16·69-s + 8·71-s − 12·73-s + 8·77-s + 8·79-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.755·7-s + 2/3·9-s − 1.20·11-s − 1.66·13-s + 0.458·19-s − 0.872·21-s − 1.66·23-s + 1.15·27-s + 0.718·31-s − 1.39·33-s − 1.92·39-s − 1.24·41-s − 0.609·43-s − 1.75·47-s + 3/7·49-s + 2.74·53-s + 0.529·57-s − 1.82·59-s + 2.30·61-s − 0.503·63-s − 0.977·67-s − 1.92·69-s + 0.949·71-s − 1.40·73-s + 0.911·77-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9588359021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9588359021\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 14 T + 162 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 18 T + 198 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 14 T + 210 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 238 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−8.201684682498111440738961744848, −8.181067651584497165084391990311, −7.45657697915533074816075617907, −7.40419844022681506264476233424, −7.11127467434369228342279222479, −6.60314583887902140125424065263, −6.15586319575175666325980060436, −5.95940051378260876274852256011, −5.23419125541024237730535293727, −5.02787559381954956328576647606, −4.77910352676138371421823941807, −4.24438535971519408445692584799, −3.64862063428608813541978481813, −3.48601634921198363027320544167, −2.85344839360193711511076587694, −2.72547172350982968307646808974, −2.13654819871845975239924236772, −2.00843366758280718083517002741, −1.06872847421987224893024490793, −0.24023489317238687848488907209,
0.24023489317238687848488907209, 1.06872847421987224893024490793, 2.00843366758280718083517002741, 2.13654819871845975239924236772, 2.72547172350982968307646808974, 2.85344839360193711511076587694, 3.48601634921198363027320544167, 3.64862063428608813541978481813, 4.24438535971519408445692584799, 4.77910352676138371421823941807, 5.02787559381954956328576647606, 5.23419125541024237730535293727, 5.95940051378260876274852256011, 6.15586319575175666325980060436, 6.60314583887902140125424065263, 7.11127467434369228342279222479, 7.40419844022681506264476233424, 7.45657697915533074816075617907, 8.181067651584497165084391990311, 8.201684682498111440738961744848
