| L(s) = 1 | + 5-s + 2·11-s − 2·13-s − 4·19-s + 2·23-s + 25-s + 6·29-s − 2·37-s + 12·41-s − 43-s + 2·47-s − 7·49-s − 8·53-s + 2·55-s + 2·59-s + 2·61-s − 2·65-s + 4·67-s + 8·71-s − 14·73-s − 8·79-s − 14·83-s + 6·89-s − 4·95-s + 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.603·11-s − 0.554·13-s − 0.917·19-s + 0.417·23-s + 1/5·25-s + 1.11·29-s − 0.328·37-s + 1.87·41-s − 0.152·43-s + 0.291·47-s − 49-s − 1.09·53-s + 0.269·55-s + 0.260·59-s + 0.256·61-s − 0.248·65-s + 0.488·67-s + 0.949·71-s − 1.63·73-s − 0.900·79-s − 1.53·83-s + 0.635·89-s − 0.410·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 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
−13.93637560529568, −13.12449120098755, −12.83393564608087, −12.46179126816699, −11.82339477043853, −11.36731844286221, −10.88927373934938, −10.29747679836601, −9.959184973377870, −9.318069908147602, −8.996905342764239, −8.408316475558934, −7.910510518121147, −7.290440028135865, −6.720264980083754, −6.373150508368704, −5.789376442711417, −5.226366847684742, −4.510880886630919, −4.285261081671732, −3.434029680605603, −2.805393305404948, −2.298362403819733, −1.571256917569071, −0.9320267347840078, 0,
0.9320267347840078, 1.571256917569071, 2.298362403819733, 2.805393305404948, 3.434029680605603, 4.285261081671732, 4.510880886630919, 5.226366847684742, 5.789376442711417, 6.373150508368704, 6.720264980083754, 7.290440028135865, 7.910510518121147, 8.408316475558934, 8.996905342764239, 9.318069908147602, 9.959184973377870, 10.29747679836601, 10.88927373934938, 11.36731844286221, 11.82339477043853, 12.46179126816699, 12.83393564608087, 13.12449120098755, 13.93637560529568
