| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s + 4·13-s − 15-s − 2·19-s − 21-s + 23-s + 25-s − 27-s − 6·29-s + 2·31-s + 35-s + 10·37-s − 4·39-s + 6·41-s + 4·43-s + 45-s + 6·47-s + 49-s + 6·53-s + 2·57-s − 12·59-s + 10·61-s + 63-s + 4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.10·13-s − 0.258·15-s − 0.458·19-s − 0.218·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.359·31-s + 0.169·35-s + 1.64·37-s − 0.640·39-s + 0.937·41-s + 0.609·43-s + 0.149·45-s + 0.875·47-s + 1/7·49-s + 0.824·53-s + 0.264·57-s − 1.56·59-s + 1.28·61-s + 0.125·63-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 154560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.204848785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.204848785\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.30804597090091, −12.85507429581766, −12.40070410191941, −11.86552988384173, −11.14226741885131, −11.08983828288294, −10.61419237623332, −10.01633325563110, −9.335092422754865, −9.196718827393909, −8.430921384562523, −7.985994520398193, −7.479954070450728, −6.849255084104229, −6.347300341755750, −5.839020213875703, −5.560635348533562, −4.875089453169360, −4.211858863426704, −3.929511801725735, −3.118376703798921, −2.372168721065352, −1.885915235161315, −1.051450185972758, −0.6404176163253709,
0.6404176163253709, 1.051450185972758, 1.885915235161315, 2.372168721065352, 3.118376703798921, 3.929511801725735, 4.211858863426704, 4.875089453169360, 5.560635348533562, 5.839020213875703, 6.347300341755750, 6.849255084104229, 7.479954070450728, 7.985994520398193, 8.430921384562523, 9.196718827393909, 9.335092422754865, 10.01633325563110, 10.61419237623332, 11.08983828288294, 11.14226741885131, 11.86552988384173, 12.40070410191941, 12.85507429581766, 13.30804597090091
