L(s) = 1 | − 3-s + 7-s + 9-s − 4·11-s − 2·13-s + 5·19-s − 21-s − 23-s − 27-s + 6·29-s − 2·31-s + 4·33-s − 2·37-s + 2·39-s + 10·41-s − 2·43-s + 3·47-s + 49-s + 2·53-s − 5·57-s − 3·59-s + 61-s + 63-s + 16·67-s + 69-s + 6·71-s + 8·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 1.14·19-s − 0.218·21-s − 0.208·23-s − 0.192·27-s + 1.11·29-s − 0.359·31-s + 0.696·33-s − 0.328·37-s + 0.320·39-s + 1.56·41-s − 0.304·43-s + 0.437·47-s + 1/7·49-s + 0.274·53-s − 0.662·57-s − 0.390·59-s + 0.128·61-s + 0.125·63-s + 1.95·67-s + 0.120·69-s + 0.712·71-s + 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.827366872\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.827366872\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−13.69941204955931, −13.38158674627047, −12.60788535061677, −12.37651256398675, −11.90394552634117, −11.17760874862595, −10.99444905702557, −10.34226698746859, −9.898517532125442, −9.491469909530755, −8.789459690271858, −8.151480137621601, −7.735463225085145, −7.293499402146711, −6.736882404528475, −6.059343200694676, −5.483217391432681, −5.092783583925717, −4.670552133436531, −3.940278078092140, −3.243242478574860, −2.544307157953196, −2.072376147400354, −1.080424485126950, −0.5037116408150421,
0.5037116408150421, 1.080424485126950, 2.072376147400354, 2.544307157953196, 3.243242478574860, 3.940278078092140, 4.670552133436531, 5.092783583925717, 5.483217391432681, 6.059343200694676, 6.736882404528475, 7.293499402146711, 7.735463225085145, 8.151480137621601, 8.789459690271858, 9.491469909530755, 9.898517532125442, 10.34226698746859, 10.99444905702557, 11.17760874862595, 11.90394552634117, 12.37651256398675, 12.60788535061677, 13.38158674627047, 13.69941204955931