| L(s) = 1 | + 3·3-s + 5-s + 6·9-s + 11-s + 2·13-s + 3·15-s + 3·17-s − 5·19-s + 3·23-s − 4·25-s + 9·27-s + 6·29-s + 3·33-s + 5·37-s + 6·39-s + 10·41-s + 4·43-s + 6·45-s − 47-s + 9·51-s + 9·53-s + 55-s − 15·57-s − 3·59-s + 3·61-s + 2·65-s + 11·67-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s + 2·9-s + 0.301·11-s + 0.554·13-s + 0.774·15-s + 0.727·17-s − 1.14·19-s + 0.625·23-s − 4/5·25-s + 1.73·27-s + 1.11·29-s + 0.522·33-s + 0.821·37-s + 0.960·39-s + 1.56·41-s + 0.609·43-s + 0.894·45-s − 0.145·47-s + 1.26·51-s + 1.23·53-s + 0.134·55-s − 1.98·57-s − 0.390·59-s + 0.384·61-s + 0.248·65-s + 1.34·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 376712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 376712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.500849754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.500849754\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.65880826701467, −12.29145479619018, −11.53658196543672, −11.02890805587123, −10.60947323447510, −9.982986403925745, −9.737820050168370, −9.200759035103254, −8.926122519082250, −8.337762043474533, −8.037073882736883, −7.660255786110235, −7.020224130867107, −6.534562534871697, −6.132394314344770, −5.487471515776094, −4.901644886357817, −4.115879435149395, −3.991747982610562, −3.422884827558299, −2.728968554549845, −2.402321576366284, −1.946222201499389, −1.171262821038813, −0.7503265201092938,
0.7503265201092938, 1.171262821038813, 1.946222201499389, 2.402321576366284, 2.728968554549845, 3.422884827558299, 3.991747982610562, 4.115879435149395, 4.901644886357817, 5.487471515776094, 6.132394314344770, 6.534562534871697, 7.020224130867107, 7.660255786110235, 8.037073882736883, 8.337762043474533, 8.926122519082250, 9.200759035103254, 9.737820050168370, 9.982986403925745, 10.60947323447510, 11.02890805587123, 11.53658196543672, 12.29145479619018, 12.65880826701467
