| L(s) = 1 | + 3-s − 2·5-s − 2·9-s − 11-s − 13-s − 2·15-s − 4·17-s + 4·19-s − 5·23-s − 25-s − 5·27-s − 6·29-s − 11·31-s − 33-s − 7·37-s − 39-s − 3·41-s − 2·43-s + 4·45-s − 3·47-s − 4·51-s + 2·53-s + 2·55-s + 4·57-s + 4·59-s + 5·61-s + 2·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 2/3·9-s − 0.301·11-s − 0.277·13-s − 0.516·15-s − 0.970·17-s + 0.917·19-s − 1.04·23-s − 1/5·25-s − 0.962·27-s − 1.11·29-s − 1.97·31-s − 0.174·33-s − 1.15·37-s − 0.160·39-s − 0.468·41-s − 0.304·43-s + 0.596·45-s − 0.437·47-s − 0.560·51-s + 0.274·53-s + 0.269·55-s + 0.529·57-s + 0.520·59-s + 0.640·61-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 11 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−15.30477085803308, −14.67191638656013, −14.40746383833007, −13.67718878528373, −13.33489492717166, −12.72380922024625, −12.03251664044641, −11.68083317323189, −11.13307845147995, −10.74385565500483, −9.848425864292021, −9.480706832882579, −8.796484288479070, −8.364968771283881, −7.891468084669893, −7.197520365926909, −7.013184125137964, −5.866829800276233, −5.532659640384605, −4.836797971925419, −3.900851050123521, −3.703920484833777, −2.964251181791903, −2.201763143399490, −1.608796994591615, 0, 0,
1.608796994591615, 2.201763143399490, 2.964251181791903, 3.703920484833777, 3.900851050123521, 4.836797971925419, 5.532659640384605, 5.866829800276233, 7.013184125137964, 7.197520365926909, 7.891468084669893, 8.364968771283881, 8.796484288479070, 9.480706832882579, 9.848425864292021, 10.74385565500483, 11.13307845147995, 11.68083317323189, 12.03251664044641, 12.72380922024625, 13.33489492717166, 13.67718878528373, 14.40746383833007, 14.67191638656013, 15.30477085803308
