| L(s) = 1 | − 2-s + 3-s + 4-s − 2·5-s − 6-s − 4·7-s − 8-s + 9-s + 2·10-s − 4·11-s + 12-s + 4·14-s − 2·15-s + 16-s + 4·17-s − 18-s − 4·19-s − 2·20-s − 4·21-s + 4·22-s + 4·23-s − 24-s − 25-s + 27-s − 4·28-s − 2·29-s + 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.20·11-s + 0.288·12-s + 1.06·14-s − 0.516·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s − 0.917·19-s − 0.447·20-s − 0.872·21-s + 0.852·22-s + 0.834·23-s − 0.204·24-s − 1/5·25-s + 0.192·27-s − 0.755·28-s − 0.371·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 167334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.34402678947622, −12.79390315079983, −12.72129620975404, −12.10954524986003, −11.49540843018333, −11.03478144841224, −10.47041042957679, −10.01417146160225, −9.745772674064728, −9.131967960594130, −8.713314473425728, −8.088228180170483, −7.685662417296394, −7.485303524252062, −6.717724749958077, −6.284041459305686, −5.816216932779438, −5.011192992009484, −4.439533959767163, −3.716688146474055, −3.282477700895604, −2.805103150266476, −2.383552209219507, −1.429369636071859, −0.5858440569720647, 0,
0.5858440569720647, 1.429369636071859, 2.383552209219507, 2.805103150266476, 3.282477700895604, 3.716688146474055, 4.439533959767163, 5.011192992009484, 5.816216932779438, 6.284041459305686, 6.717724749958077, 7.485303524252062, 7.685662417296394, 8.088228180170483, 8.713314473425728, 9.131967960594130, 9.745772674064728, 10.01417146160225, 10.47041042957679, 11.03478144841224, 11.49540843018333, 12.10954524986003, 12.72129620975404, 12.79390315079983, 13.34402678947622
