| L(s) = 1 | − 2-s + 4-s − 2·5-s − 8-s + 2·10-s − 4·11-s + 2·13-s + 16-s + 17-s − 4·19-s − 2·20-s + 4·22-s − 8·23-s − 25-s − 2·26-s − 6·29-s − 32-s − 34-s − 2·37-s + 4·38-s + 2·40-s + 10·41-s − 4·43-s − 4·44-s + 8·46-s + 50-s + 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s + 0.632·10-s − 1.20·11-s + 0.554·13-s + 1/4·16-s + 0.242·17-s − 0.917·19-s − 0.447·20-s + 0.852·22-s − 1.66·23-s − 1/5·25-s − 0.392·26-s − 1.11·29-s − 0.176·32-s − 0.171·34-s − 0.328·37-s + 0.648·38-s + 0.316·40-s + 1.56·41-s − 0.609·43-s − 0.603·44-s + 1.17·46-s + 0.141·50-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14994 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14994 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2738939324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2738939324\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.02752776320075, −15.68440287983745, −15.12220770200554, −14.58694823526123, −13.76869861271637, −13.26597539207187, −12.40466704885488, −12.25938897177796, −11.30008228493457, −10.99557356274660, −10.43182474758004, −9.787226661541050, −9.189797119669579, −8.335080239993701, −8.015210985031560, −7.607676668611571, −6.868331950798567, −5.992492893527649, −5.623994701626662, −4.534453369361329, −3.958432430747003, −3.208682621087435, −2.343874970756038, −1.581856066938266, −0.2480413364873831,
0.2480413364873831, 1.581856066938266, 2.343874970756038, 3.208682621087435, 3.958432430747003, 4.534453369361329, 5.623994701626662, 5.992492893527649, 6.868331950798567, 7.607676668611571, 8.015210985031560, 8.335080239993701, 9.189797119669579, 9.787226661541050, 10.43182474758004, 10.99557356274660, 11.30008228493457, 12.25938897177796, 12.40466704885488, 13.26597539207187, 13.76869861271637, 14.58694823526123, 15.12220770200554, 15.68440287983745, 16.02752776320075
