| L(s) = 1 | − 2·4-s + 2·5-s − 4·11-s − 6·13-s + 4·17-s − 2·19-s − 4·20-s + 4·23-s + 3·25-s − 8·29-s − 6·31-s − 14·37-s + 4·41-s − 18·43-s + 8·44-s + 4·47-s + 12·52-s − 8·53-s − 8·55-s + 8·61-s + 8·64-s − 12·65-s − 2·67-s − 8·68-s + 4·71-s − 10·73-s + 4·76-s + ⋯ |
| L(s) = 1 | − 4-s + 0.894·5-s − 1.20·11-s − 1.66·13-s + 0.970·17-s − 0.458·19-s − 0.894·20-s + 0.834·23-s + 3/5·25-s − 1.48·29-s − 1.07·31-s − 2.30·37-s + 0.624·41-s − 2.74·43-s + 1.20·44-s + 0.583·47-s + 1.66·52-s − 1.09·53-s − 1.07·55-s + 1.02·61-s + 64-s − 1.48·65-s − 0.244·67-s − 0.970·68-s + 0.474·71-s − 1.17·73-s + 0.458·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 33 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 32 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 56 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 63 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 14 T + 121 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 68 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 18 T + 165 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T - 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 114 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 116 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 144 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 121 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 127 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 180 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 16 T + 224 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 250 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−8.929686244154819305558289411022, −8.488887111355729783119856310938, −8.228016399217033639889557699453, −7.65441333948184578090468669640, −7.27371705222410705166812738190, −7.07169630563717543805100391490, −6.57624215580908582948400558405, −5.97409721945613783302716220973, −5.40340311266317447282884110795, −5.29077956190001027215657897974, −4.92903632996141049890689475160, −4.73714424405971593158478407265, −3.80778039628749942844786001024, −3.60266820088148661128152648561, −2.91536653365491275541117719275, −2.46056834043268076429489661238, −1.92271414774413219832214593795, −1.38193783667846728910886421957, 0, 0,
1.38193783667846728910886421957, 1.92271414774413219832214593795, 2.46056834043268076429489661238, 2.91536653365491275541117719275, 3.60266820088148661128152648561, 3.80778039628749942844786001024, 4.73714424405971593158478407265, 4.92903632996141049890689475160, 5.29077956190001027215657897974, 5.40340311266317447282884110795, 5.97409721945613783302716220973, 6.57624215580908582948400558405, 7.07169630563717543805100391490, 7.27371705222410705166812738190, 7.65441333948184578090468669640, 8.228016399217033639889557699453, 8.488887111355729783119856310938, 8.929686244154819305558289411022
