L(s) = 1 | + 5-s − 7-s + 5·11-s + 13-s − 3·17-s − 6·19-s + 6·23-s + 25-s + 9·29-s − 35-s + 6·37-s − 8·41-s + 6·43-s − 3·47-s + 49-s + 12·53-s + 5·55-s − 8·59-s − 4·61-s + 65-s − 4·67-s − 8·71-s + 10·73-s − 5·77-s − 3·79-s + 12·83-s − 3·85-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s + 1.50·11-s + 0.277·13-s − 0.727·17-s − 1.37·19-s + 1.25·23-s + 1/5·25-s + 1.67·29-s − 0.169·35-s + 0.986·37-s − 1.24·41-s + 0.914·43-s − 0.437·47-s + 1/7·49-s + 1.64·53-s + 0.674·55-s − 1.04·59-s − 0.512·61-s + 0.124·65-s − 0.488·67-s − 0.949·71-s + 1.17·73-s − 0.569·77-s − 0.337·79-s + 1.31·83-s − 0.325·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.069710472\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.069710472\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−8.925232497758383288239481694498, −8.418272725763958407124348894851, −7.16216561796569421885219307291, −6.45877795748859351665331711034, −6.14206553924090701620723770176, −4.83171016164985859832688933282, −4.17514659999582833871651190557, −3.16093778472040229481013774454, −2.11101541760584806136071858440, −0.949618884734753047450408637076,
0.949618884734753047450408637076, 2.11101541760584806136071858440, 3.16093778472040229481013774454, 4.17514659999582833871651190557, 4.83171016164985859832688933282, 6.14206553924090701620723770176, 6.45877795748859351665331711034, 7.16216561796569421885219307291, 8.418272725763958407124348894851, 8.925232497758383288239481694498