| L(s) = 1 | − 3-s − 5-s − 2·9-s + 2·11-s + 4·13-s + 15-s + 6·19-s + 3·23-s + 25-s + 5·27-s + 3·29-s − 2·33-s + 12·37-s − 4·39-s + 7·41-s + 9·43-s + 2·45-s + 6·53-s − 2·55-s − 6·57-s − 10·59-s + 5·61-s − 4·65-s − 11·67-s − 3·69-s − 10·71-s + 8·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 2/3·9-s + 0.603·11-s + 1.10·13-s + 0.258·15-s + 1.37·19-s + 0.625·23-s + 1/5·25-s + 0.962·27-s + 0.557·29-s − 0.348·33-s + 1.97·37-s − 0.640·39-s + 1.09·41-s + 1.37·43-s + 0.298·45-s + 0.824·53-s − 0.269·55-s − 0.794·57-s − 1.30·59-s + 0.640·61-s − 0.496·65-s − 1.34·67-s − 0.361·69-s − 1.18·71-s + 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.930481768\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.930481768\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 17 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−16.04630752123941, −15.60899303073951, −14.83098623691399, −14.35847813010524, −13.79922324029524, −13.23690526311393, −12.52213325992671, −11.93303886203286, −11.51814369428591, −10.98287936813612, −10.64091741710147, −9.600254345793332, −9.189969868172366, −8.554212621869222, −7.897145348461680, −7.306176459786158, −6.555625793091157, −5.915990565476152, −5.559739820464971, −4.594790261054093, −4.070146875333812, −3.182161854066597, −2.653674206690180, −1.238683050159451, −0.7329365122804462,
0.7329365122804462, 1.238683050159451, 2.653674206690180, 3.182161854066597, 4.070146875333812, 4.594790261054093, 5.559739820464971, 5.915990565476152, 6.555625793091157, 7.306176459786158, 7.897145348461680, 8.554212621869222, 9.189969868172366, 9.600254345793332, 10.64091741710147, 10.98287936813612, 11.51814369428591, 11.93303886203286, 12.52213325992671, 13.23690526311393, 13.79922324029524, 14.35847813010524, 14.83098623691399, 15.60899303073951, 16.04630752123941
