L(s) = 1 | + 5-s − 7-s − 2·11-s − 6·13-s + 2·17-s + 4·19-s + 2·23-s + 25-s + 4·31-s − 35-s + 10·37-s − 10·41-s + 12·43-s + 12·47-s + 49-s + 4·53-s − 2·55-s + 4·59-s + 6·61-s − 6·65-s − 4·67-s − 2·71-s + 6·73-s + 2·77-s + 8·79-s + 16·83-s + 2·85-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s − 0.603·11-s − 1.66·13-s + 0.485·17-s + 0.917·19-s + 0.417·23-s + 1/5·25-s + 0.718·31-s − 0.169·35-s + 1.64·37-s − 1.56·41-s + 1.82·43-s + 1.75·47-s + 1/7·49-s + 0.549·53-s − 0.269·55-s + 0.520·59-s + 0.768·61-s − 0.744·65-s − 0.488·67-s − 0.237·71-s + 0.702·73-s + 0.227·77-s + 0.900·79-s + 1.75·83-s + 0.216·85-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
\(L(1)\) |
\(\approx\) |
\(1.678186364\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.678186364\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−9.114129994262862706037357444743, −7.992486128823607112421964855532, −7.41755978021551121899363591900, −6.69038649030359876405958103618, −5.63540628159815430842762962298, −5.14630449312891439374894043808, −4.16769579482428738569986317211, −2.91810588791655844201446652698, −2.36814983720652614384320986608, −0.811430369368519426527923768467,
0.811430369368519426527923768467, 2.36814983720652614384320986608, 2.91810588791655844201446652698, 4.16769579482428738569986317211, 5.14630449312891439374894043808, 5.63540628159815430842762962298, 6.69038649030359876405958103618, 7.41755978021551121899363591900, 7.992486128823607112421964855532, 9.114129994262862706037357444743