L(s) = 1 | − 3-s + 4·7-s + 9-s − 2·11-s + 13-s + 6·17-s + 4·19-s − 4·21-s − 4·23-s − 27-s − 6·29-s + 8·31-s + 2·33-s + 10·37-s − 39-s − 4·41-s + 4·43-s + 6·47-s + 9·49-s − 6·51-s − 6·53-s − 4·57-s − 6·59-s − 6·61-s + 4·63-s + 4·69-s + 10·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s − 0.603·11-s + 0.277·13-s + 1.45·17-s + 0.917·19-s − 0.872·21-s − 0.834·23-s − 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.348·33-s + 1.64·37-s − 0.160·39-s − 0.624·41-s + 0.609·43-s + 0.875·47-s + 9/7·49-s − 0.840·51-s − 0.824·53-s − 0.529·57-s − 0.781·59-s − 0.768·61-s + 0.503·63-s + 0.481·69-s + 1.18·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.241154783\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.241154783\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.84962745364494199681346233099, −7.43888987414538512625565077892, −6.28645018760350274632513809095, −5.67279599652778019751508607751, −5.11302317295690314450266998360, −4.48744327024597533271485901775, −3.62719901488303768637548355448, −2.60061188970266576039715768575, −1.58843544568197716308506532402, −0.833742737890796387559892207253,
0.833742737890796387559892207253, 1.58843544568197716308506532402, 2.60061188970266576039715768575, 3.62719901488303768637548355448, 4.48744327024597533271485901775, 5.11302317295690314450266998360, 5.67279599652778019751508607751, 6.28645018760350274632513809095, 7.43888987414538512625565077892, 7.84962745364494199681346233099