| L(s) = 1 | − 5-s − 11-s + 6·13-s − 3·17-s + 6·19-s − 23-s − 4·25-s − 10·29-s + 8·31-s − 2·37-s − 5·41-s + 6·43-s − 3·47-s + 6·53-s + 55-s − 8·59-s − 9·61-s − 6·65-s + 13·67-s − 12·71-s − 4·73-s + 5·79-s + 15·83-s + 3·85-s + 18·89-s − 6·95-s − 9·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s + 1.66·13-s − 0.727·17-s + 1.37·19-s − 0.208·23-s − 4/5·25-s − 1.85·29-s + 1.43·31-s − 0.328·37-s − 0.780·41-s + 0.914·43-s − 0.437·47-s + 0.824·53-s + 0.134·55-s − 1.04·59-s − 1.15·61-s − 0.744·65-s + 1.58·67-s − 1.42·71-s − 0.468·73-s + 0.562·79-s + 1.64·83-s + 0.325·85-s + 1.90·89-s − 0.615·95-s − 0.913·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−15.26398425224507, −14.63109312178187, −13.75410189858345, −13.56527976107113, −13.25008320913011, −12.38081763922895, −11.87764787274937, −11.41001167478824, −10.96898462451958, −10.44347348906438, −9.762166172432881, −9.157918640311914, −8.757229518507583, −7.956036826377761, −7.740722863528215, −7.016292841399896, −6.283810928475222, −5.881288139823684, −5.199557275408612, −4.522475366127309, −3.708463363683013, −3.511279535232494, −2.581960522259632, −1.736190824880391, −1.007160933946653, 0,
1.007160933946653, 1.736190824880391, 2.581960522259632, 3.511279535232494, 3.708463363683013, 4.522475366127309, 5.199557275408612, 5.881288139823684, 6.283810928475222, 7.016292841399896, 7.740722863528215, 7.956036826377761, 8.757229518507583, 9.157918640311914, 9.762166172432881, 10.44347348906438, 10.96898462451958, 11.41001167478824, 11.87764787274937, 12.38081763922895, 13.25008320913011, 13.56527976107113, 13.75410189858345, 14.63109312178187, 15.26398425224507
