| L(s) = 1 | − 3-s − 5-s − 4·7-s + 9-s − 2·13-s + 15-s − 4·19-s + 4·21-s + 25-s − 27-s − 2·29-s − 10·31-s + 4·35-s + 10·37-s + 2·39-s − 6·41-s − 12·43-s − 45-s − 6·47-s + 9·49-s + 12·53-s + 4·57-s + 12·59-s + 8·61-s − 4·63-s + 2·65-s − 8·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.554·13-s + 0.258·15-s − 0.917·19-s + 0.872·21-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 1.79·31-s + 0.676·35-s + 1.64·37-s + 0.320·39-s − 0.937·41-s − 1.82·43-s − 0.149·45-s − 0.875·47-s + 9/7·49-s + 1.64·53-s + 0.529·57-s + 1.56·59-s + 1.02·61-s − 0.503·63-s + 0.248·65-s − 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−14.66215246417236, −13.67504122588402, −13.22237891288116, −12.88788524818827, −12.46529851910061, −11.93717166781056, −11.37777241583555, −10.96344130192702, −10.29851381628911, −9.765197663122597, −9.611793179492992, −8.723726796043708, −8.380199743054326, −7.530115001818864, −7.049569635812454, −6.639681677580560, −6.161521950553261, −5.443697682369514, −5.059137987347134, −4.108617668693572, −3.820672174588508, −3.131995165136124, −2.442858106700009, −1.696793296782448, −0.5769452518079123, 0,
0.5769452518079123, 1.696793296782448, 2.442858106700009, 3.131995165136124, 3.820672174588508, 4.108617668693572, 5.059137987347134, 5.443697682369514, 6.161521950553261, 6.639681677580560, 7.049569635812454, 7.530115001818864, 8.380199743054326, 8.723726796043708, 9.611793179492992, 9.765197663122597, 10.29851381628911, 10.96344130192702, 11.37777241583555, 11.93717166781056, 12.46529851910061, 12.88788524818827, 13.22237891288116, 13.67504122588402, 14.66215246417236
