| L(s) = 1 | + 3-s + 3·5-s + 2·7-s − 2·9-s + 2·11-s + 8·13-s + 3·15-s + 2·17-s + 2·19-s + 2·21-s + 12·23-s − 2·27-s + 10·29-s − 31-s + 2·33-s + 6·35-s + 2·37-s + 8·39-s + 5·41-s − 9·43-s − 6·45-s − 4·47-s + 3·49-s + 2·51-s + 19·53-s + 6·55-s + 2·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s + 0.755·7-s − 2/3·9-s + 0.603·11-s + 2.21·13-s + 0.774·15-s + 0.485·17-s + 0.458·19-s + 0.436·21-s + 2.50·23-s − 0.384·27-s + 1.85·29-s − 0.179·31-s + 0.348·33-s + 1.01·35-s + 0.328·37-s + 1.28·39-s + 0.780·41-s − 1.37·43-s − 0.894·45-s − 0.583·47-s + 3/7·49-s + 0.280·51-s + 2.60·53-s + 0.809·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58003456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58003456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.09797162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.09797162\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 3 T + 9 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 26 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 70 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + 33 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 62 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 5 T + 85 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 9 T + 103 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 19 T + 193 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 13 T + 161 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 9 T + 73 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 138 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + T + 117 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 142 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 20 T + 226 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 5 T + 197 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
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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
−8.145117245576285035611847798654, −7.895558826599912892279585815090, −7.20408426425167702053093679898, −7.04577750145546687730663919234, −6.59639545805124625488555263426, −6.30661371649266310320742525485, −5.97224574128897542816915031891, −5.55238750926577033356981973439, −5.34957690577874002576678040670, −5.05190671998961460900172891739, −4.45180787246456566322049477461, −4.13618802409150009926181298607, −3.53617164722893198025667669457, −3.36622449207444284263608943835, −2.84722384955827859708488149548, −2.52108343743143549413464637703, −2.04707138144833725594222833521, −1.40385363344112298267582649701, −1.12312830830933072074763878498, −0.852598322904651585987461620056,
0.852598322904651585987461620056, 1.12312830830933072074763878498, 1.40385363344112298267582649701, 2.04707138144833725594222833521, 2.52108343743143549413464637703, 2.84722384955827859708488149548, 3.36622449207444284263608943835, 3.53617164722893198025667669457, 4.13618802409150009926181298607, 4.45180787246456566322049477461, 5.05190671998961460900172891739, 5.34957690577874002576678040670, 5.55238750926577033356981973439, 5.97224574128897542816915031891, 6.30661371649266310320742525485, 6.59639545805124625488555263426, 7.04577750145546687730663919234, 7.20408426425167702053093679898, 7.895558826599912892279585815090, 8.145117245576285035611847798654
