L(s) = 1 | + 3-s + 7-s + 9-s + 11-s − 6·13-s − 2·17-s + 4·19-s + 21-s − 8·23-s + 27-s − 2·29-s + 8·31-s + 33-s − 6·37-s − 6·39-s − 6·41-s + 4·43-s − 8·47-s + 49-s − 2·51-s − 6·53-s + 4·57-s − 12·59-s + 14·61-s + 63-s + 4·67-s − 8·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.66·13-s − 0.485·17-s + 0.917·19-s + 0.218·21-s − 1.66·23-s + 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.174·33-s − 0.986·37-s − 0.960·39-s − 0.937·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.280·51-s − 0.824·53-s + 0.529·57-s − 1.56·59-s + 1.79·61-s + 0.125·63-s + 0.488·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.142111128\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.142111128\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.45659343571950, −14.11106299709188, −13.82572183232601, −13.11773309425128, −12.47516686839878, −12.02364269217096, −11.68281448711750, −11.03473383715876, −10.17619467276150, −9.935275134438182, −9.473080563802114, −8.814670746926411, −8.189305825746647, −7.773474296169153, −7.283838808330422, −6.621246683869787, −6.106576331836975, −5.096646085114400, −4.919369982353503, −4.138005006522920, −3.512735403920236, −2.789949918487132, −2.134851550233491, −1.613244042645747, −0.4820101234040509,
0.4820101234040509, 1.613244042645747, 2.134851550233491, 2.789949918487132, 3.512735403920236, 4.138005006522920, 4.919369982353503, 5.096646085114400, 6.106576331836975, 6.621246683869787, 7.283838808330422, 7.773474296169153, 8.189305825746647, 8.814670746926411, 9.473080563802114, 9.935275134438182, 10.17619467276150, 11.03473383715876, 11.68281448711750, 12.02364269217096, 12.47516686839878, 13.11773309425128, 13.82572183232601, 14.11106299709188, 14.45659343571950