| L(s) = 1 | − 5-s − 5·7-s − 11-s − 2·17-s − 6·19-s + 5·23-s + 25-s − 5·29-s − 3·31-s + 5·35-s − 6·37-s + 7·41-s + 7·43-s + 3·47-s + 18·49-s + 11·53-s + 55-s + 10·59-s − 10·61-s − 2·71-s + 7·73-s + 5·77-s + 12·79-s + 8·83-s + 2·85-s + 18·89-s + 6·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.88·7-s − 0.301·11-s − 0.485·17-s − 1.37·19-s + 1.04·23-s + 1/5·25-s − 0.928·29-s − 0.538·31-s + 0.845·35-s − 0.986·37-s + 1.09·41-s + 1.06·43-s + 0.437·47-s + 18/7·49-s + 1.51·53-s + 0.134·55-s + 1.30·59-s − 1.28·61-s − 0.237·71-s + 0.819·73-s + 0.569·77-s + 1.35·79-s + 0.878·83-s + 0.216·85-s + 1.90·89-s + 0.615·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.282105823\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.282105823\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−12.68535577613576, −12.25081776974335, −11.82997772795246, −10.96224222176515, −10.80553555181308, −10.46472112540850, −9.788596732548708, −9.331817196226139, −8.955856795074860, −8.681486973341758, −7.933632089138930, −7.339932038282560, −7.050445912238028, −6.562275021160404, −6.116777769511760, −5.639043429336420, −5.092651618739892, −4.375815547187982, −3.852529023505463, −3.590577662409892, −2.899980732838270, −2.428538667926192, −1.926415403025015, −0.7544864132729367, −0.4090319842164208,
0.4090319842164208, 0.7544864132729367, 1.926415403025015, 2.428538667926192, 2.899980732838270, 3.590577662409892, 3.852529023505463, 4.375815547187982, 5.092651618739892, 5.639043429336420, 6.116777769511760, 6.562275021160404, 7.050445912238028, 7.339932038282560, 7.933632089138930, 8.681486973341758, 8.955856795074860, 9.331817196226139, 9.788596732548708, 10.46472112540850, 10.80553555181308, 10.96224222176515, 11.82997772795246, 12.25081776974335, 12.68535577613576
