| L(s) = 1 | − 2-s + 4-s − 8-s − 3·11-s + 16-s + 6·17-s + 3·22-s − 9·23-s + 7·29-s + 31-s − 32-s − 6·34-s + 37-s − 13·43-s − 3·44-s + 9·46-s − 11·47-s − 7·49-s + 6·53-s − 7·58-s + 11·59-s − 62-s + 64-s + 4·67-s + 6·68-s − 4·71-s − 74-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 0.904·11-s + 1/4·16-s + 1.45·17-s + 0.639·22-s − 1.87·23-s + 1.29·29-s + 0.179·31-s − 0.176·32-s − 1.02·34-s + 0.164·37-s − 1.98·43-s − 0.452·44-s + 1.32·46-s − 1.60·47-s − 49-s + 0.824·53-s − 0.919·58-s + 1.43·59-s − 0.127·62-s + 1/8·64-s + 0.488·67-s + 0.727·68-s − 0.474·71-s − 0.116·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−14.34746598035797, −13.89707976541083, −13.18884447009564, −12.83624790563156, −12.13477866396542, −11.72118448676131, −11.45862325186068, −10.48093997901981, −10.18139153876737, −9.953389156874094, −9.395646074621241, −8.466674376723417, −8.173841054710248, −7.929138506849296, −7.205559169657000, −6.598711220573948, −6.108674174960144, −5.457170395008342, −5.005162202885615, −4.260900430290400, −3.419393092930706, −3.056363151051717, −2.217954542160367, −1.663133390596103, −0.8094137608646691, 0,
0.8094137608646691, 1.663133390596103, 2.217954542160367, 3.056363151051717, 3.419393092930706, 4.260900430290400, 5.005162202885615, 5.457170395008342, 6.108674174960144, 6.598711220573948, 7.205559169657000, 7.929138506849296, 8.173841054710248, 8.466674376723417, 9.395646074621241, 9.953389156874094, 10.18139153876737, 10.48093997901981, 11.45862325186068, 11.72118448676131, 12.13477866396542, 12.83624790563156, 13.18884447009564, 13.89707976541083, 14.34746598035797
