| L(s) = 1 | + 2·5-s + 2·7-s − 4·11-s + 4·17-s − 10·19-s + 2·23-s − 25-s − 4·29-s − 12·31-s + 4·35-s + 4·37-s − 4·43-s + 3·49-s − 12·53-s − 8·55-s + 4·59-s + 18·61-s + 16·67-s − 10·71-s − 4·73-s − 8·77-s − 6·79-s − 4·83-s + 8·85-s − 24·89-s − 20·95-s − 4·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s − 1.20·11-s + 0.970·17-s − 2.29·19-s + 0.417·23-s − 1/5·25-s − 0.742·29-s − 2.15·31-s + 0.676·35-s + 0.657·37-s − 0.609·43-s + 3/7·49-s − 1.64·53-s − 1.07·55-s + 0.520·59-s + 2.30·61-s + 1.95·67-s − 1.18·71-s − 0.468·73-s − 0.911·77-s − 0.675·79-s − 0.439·83-s + 0.867·85-s − 2.54·89-s − 2.05·95-s − 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82301184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82301184 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 10 T + 3 p T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 18 T + 197 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 10 T + 143 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 143 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 24 T + 298 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 174 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.58761710652720195156742051614, −7.19214790061839753536400566553, −6.79224618369245653425331874532, −6.73298400118731896379801246875, −5.92755781852134525044680218058, −5.87626955103663284117943255052, −5.43158895735267705839239257692, −5.33413601353287708020568883724, −4.86116285514215150893290427069, −4.46078741025801186572540884345, −3.90146075078856934241532755370, −3.86268491105692009400289026148, −3.19817243037324302017028978035, −2.69907922308735480980961234476, −2.23555422889098530313427657163, −2.14403024617708486648619532865, −1.48914637440853416430666522655, −1.22321944397217321427649375480, 0, 0,
1.22321944397217321427649375480, 1.48914637440853416430666522655, 2.14403024617708486648619532865, 2.23555422889098530313427657163, 2.69907922308735480980961234476, 3.19817243037324302017028978035, 3.86268491105692009400289026148, 3.90146075078856934241532755370, 4.46078741025801186572540884345, 4.86116285514215150893290427069, 5.33413601353287708020568883724, 5.43158895735267705839239257692, 5.87626955103663284117943255052, 5.92755781852134525044680218058, 6.73298400118731896379801246875, 6.79224618369245653425331874532, 7.19214790061839753536400566553, 7.58761710652720195156742051614
