| L(s) = 1 | + 2·5-s + 2·7-s + 4·13-s + 6·17-s + 2·19-s + 3·25-s + 12·29-s + 2·31-s + 4·35-s − 2·37-s + 12·41-s − 10·43-s − 6·47-s − 8·49-s + 18·53-s − 12·59-s − 8·61-s + 8·65-s + 8·67-s − 12·71-s + 10·73-s + 8·79-s − 6·83-s + 12·85-s + 8·91-s + 4·95-s − 2·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s + 1.10·13-s + 1.45·17-s + 0.458·19-s + 3/5·25-s + 2.22·29-s + 0.359·31-s + 0.676·35-s − 0.328·37-s + 1.87·41-s − 1.52·43-s − 0.875·47-s − 8/7·49-s + 2.47·53-s − 1.56·59-s − 1.02·61-s + 0.992·65-s + 0.977·67-s − 1.42·71-s + 1.17·73-s + 0.900·79-s − 0.658·83-s + 1.30·85-s + 0.838·91-s + 0.410·95-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41990400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41990400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.036571764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.036571764\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 40 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 48 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 108 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 18 T + 184 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 151 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 151 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 10 T + 144 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 151 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 192 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−8.146918167600598776108243481729, −8.022175519126736491804318345645, −7.38412984418221750287415715713, −7.29933172394859758977834813784, −6.57701673419487594083303580603, −6.46533270882855706074519438040, −6.05230079863138445783003937212, −5.77589532330326647914965185883, −5.29434939869546583564258902035, −5.08307898543234930119941655178, −4.55607684441143564552200608143, −4.43330748845971855041970699839, −3.57635828465849703022253879022, −3.52165234018459590788960762004, −2.83201397202957202783191456119, −2.69163737490376644624863095695, −1.95117033731855829177907842489, −1.47216118033436652737423148887, −1.16534587693512391321881678275, −0.67944127409426459877210398832,
0.67944127409426459877210398832, 1.16534587693512391321881678275, 1.47216118033436652737423148887, 1.95117033731855829177907842489, 2.69163737490376644624863095695, 2.83201397202957202783191456119, 3.52165234018459590788960762004, 3.57635828465849703022253879022, 4.43330748845971855041970699839, 4.55607684441143564552200608143, 5.08307898543234930119941655178, 5.29434939869546583564258902035, 5.77589532330326647914965185883, 6.05230079863138445783003937212, 6.46533270882855706074519438040, 6.57701673419487594083303580603, 7.29933172394859758977834813784, 7.38412984418221750287415715713, 8.022175519126736491804318345645, 8.146918167600598776108243481729
