| L(s) = 1 | − 2·4-s + 5-s − 2·7-s + 11-s − 13-s + 4·16-s + 4·17-s + 2·19-s − 2·20-s − 7·23-s − 4·25-s + 4·28-s + 2·29-s − 3·31-s − 2·35-s − 11·37-s − 10·41-s − 4·43-s − 2·44-s + 4·47-s − 3·49-s + 2·52-s − 2·53-s + 55-s + 59-s − 2·61-s − 8·64-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s − 0.755·7-s + 0.301·11-s − 0.277·13-s + 16-s + 0.970·17-s + 0.458·19-s − 0.447·20-s − 1.45·23-s − 4/5·25-s + 0.755·28-s + 0.371·29-s − 0.538·31-s − 0.338·35-s − 1.80·37-s − 1.56·41-s − 0.609·43-s − 0.301·44-s + 0.583·47-s − 3/7·49-s + 0.277·52-s − 0.274·53-s + 0.134·55-s + 0.130·59-s − 0.256·61-s − 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−9.439614795403392159339706573644, −8.528601897910997907766991611046, −7.76116009742030896542487448970, −6.70861289370917133535141316633, −5.76971326436299567396516934204, −5.13164450675750106555447729059, −3.93044483272471932820766011024, −3.23573160090042234483516934690, −1.66131996401817111295419273715, 0,
1.66131996401817111295419273715, 3.23573160090042234483516934690, 3.93044483272471932820766011024, 5.13164450675750106555447729059, 5.76971326436299567396516934204, 6.70861289370917133535141316633, 7.76116009742030896542487448970, 8.528601897910997907766991611046, 9.439614795403392159339706573644
