L(s) = 1 | − 3-s + 9-s + 11-s + 2·13-s + 6·17-s + 4·19-s − 27-s + 6·29-s + 8·31-s − 33-s − 6·37-s − 2·39-s + 10·41-s − 4·43-s − 8·47-s − 7·49-s − 6·51-s + 10·53-s − 4·57-s + 12·59-s + 6·61-s − 4·67-s + 14·73-s + 81-s + 4·83-s − 6·87-s − 6·89-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.301·11-s + 0.554·13-s + 1.45·17-s + 0.917·19-s − 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.174·33-s − 0.986·37-s − 0.320·39-s + 1.56·41-s − 0.609·43-s − 1.16·47-s − 49-s − 0.840·51-s + 1.37·53-s − 0.529·57-s + 1.56·59-s + 0.768·61-s − 0.488·67-s + 1.63·73-s + 1/9·81-s + 0.439·83-s − 0.643·87-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.253592398\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.253592398\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−16.23515356776139, −15.85994025484203, −15.19286087946564, −14.42858132138331, −14.06932161665215, −13.41815109441764, −12.79139627814599, −12.09389449698874, −11.77413910581588, −11.23422359554211, −10.39183616679602, −10.00355036625704, −9.480411825581867, −8.568854295944422, −8.083019212052405, −7.399543407923355, −6.639648623816088, −6.191103447786898, −5.351228117004448, −4.977088542828107, −3.992801458045159, −3.390037660664842, −2.559158806498639, −1.338528014430990, −0.7990863200188872,
0.7990863200188872, 1.338528014430990, 2.559158806498639, 3.390037660664842, 3.992801458045159, 4.977088542828107, 5.351228117004448, 6.191103447786898, 6.639648623816088, 7.399543407923355, 8.083019212052405, 8.568854295944422, 9.480411825581867, 10.00355036625704, 10.39183616679602, 11.23422359554211, 11.77413910581588, 12.09389449698874, 12.79139627814599, 13.41815109441764, 14.06932161665215, 14.42858132138331, 15.19286087946564, 15.85994025484203, 16.23515356776139