L(s) = 1 | + 3-s + 7-s + 9-s − 5·11-s − 13-s + 21-s + 6·23-s + 27-s − 2·29-s − 31-s − 5·33-s + 7·37-s − 39-s − 9·41-s + 11·43-s + 3·47-s − 6·49-s + 13·53-s + 61-s + 63-s + 4·67-s + 6·69-s − 71-s + 4·73-s − 5·77-s − 10·79-s + 81-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.50·11-s − 0.277·13-s + 0.218·21-s + 1.25·23-s + 0.192·27-s − 0.371·29-s − 0.179·31-s − 0.870·33-s + 1.15·37-s − 0.160·39-s − 1.40·41-s + 1.67·43-s + 0.437·47-s − 6/7·49-s + 1.78·53-s + 0.128·61-s + 0.125·63-s + 0.488·67-s + 0.722·69-s − 0.118·71-s + 0.468·73-s − 0.569·77-s − 1.12·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.526808916\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.526808916\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.84384482323942, −14.49811531893327, −13.74971333354928, −13.25319470929425, −12.97109897587741, −12.38061843939627, −11.71149903589984, −11.07209853467669, −10.66277383178800, −10.12370363934900, −9.520256369565638, −8.981562820104905, −8.358017486514600, −7.928091008578437, −7.332589250631945, −6.966686618208065, −6.037029607859267, −5.361956265662251, −4.969116255176937, −4.261261952753965, −3.552504036621829, −2.687900978176729, −2.471244836384722, −1.496297700015390, −0.5663570578734877,
0.5663570578734877, 1.496297700015390, 2.471244836384722, 2.687900978176729, 3.552504036621829, 4.261261952753965, 4.969116255176937, 5.361956265662251, 6.037029607859267, 6.966686618208065, 7.332589250631945, 7.928091008578437, 8.358017486514600, 8.981562820104905, 9.520256369565638, 10.12370363934900, 10.66277383178800, 11.07209853467669, 11.71149903589984, 12.38061843939627, 12.97109897587741, 13.25319470929425, 13.74971333354928, 14.49811531893327, 14.84384482323942