L(s) = 1 | + 5-s + 4·7-s − 4·11-s + 6·13-s + 2·17-s + 4·19-s − 23-s + 25-s + 2·29-s + 4·35-s + 2·37-s − 2·41-s + 9·49-s + 6·53-s − 4·55-s + 8·59-s − 2·61-s + 6·65-s − 16·67-s + 4·71-s + 10·73-s − 16·77-s − 4·83-s + 2·85-s − 6·89-s + 24·91-s + 4·95-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.51·7-s − 1.20·11-s + 1.66·13-s + 0.485·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s + 0.371·29-s + 0.676·35-s + 0.328·37-s − 0.312·41-s + 9/7·49-s + 0.824·53-s − 0.539·55-s + 1.04·59-s − 0.256·61-s + 0.744·65-s − 1.95·67-s + 0.474·71-s + 1.17·73-s − 1.82·77-s − 0.439·83-s + 0.216·85-s − 0.635·89-s + 2.51·91-s + 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.132255932\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.132255932\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.895908578706044276541732863707, −7.29936423092044665287347241010, −6.30167121334597291857036751593, −5.56490917809100656125096643835, −5.18608418173652300323324898321, −4.36691986594063349208128250708, −3.48008440775626251414324873888, −2.58761177646133317525090584817, −1.66327876536016891391595409133, −0.950408983298995558454951528614,
0.950408983298995558454951528614, 1.66327876536016891391595409133, 2.58761177646133317525090584817, 3.48008440775626251414324873888, 4.36691986594063349208128250708, 5.18608418173652300323324898321, 5.56490917809100656125096643835, 6.30167121334597291857036751593, 7.29936423092044665287347241010, 7.895908578706044276541732863707