| L(s) = 1 | + 2-s + 4-s + 8-s + 11-s + 2·13-s + 16-s − 2·17-s − 4·19-s + 22-s − 4·23-s + 2·26-s + 6·29-s + 32-s − 2·34-s − 2·37-s − 4·38-s + 6·41-s − 12·43-s + 44-s − 4·46-s − 8·47-s + 2·52-s − 6·53-s + 6·58-s + 8·59-s + 14·61-s + 64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 0.301·11-s + 0.554·13-s + 1/4·16-s − 0.485·17-s − 0.917·19-s + 0.213·22-s − 0.834·23-s + 0.392·26-s + 1.11·29-s + 0.176·32-s − 0.342·34-s − 0.328·37-s − 0.648·38-s + 0.937·41-s − 1.82·43-s + 0.150·44-s − 0.589·46-s − 1.16·47-s + 0.277·52-s − 0.824·53-s + 0.787·58-s + 1.04·59-s + 1.79·61-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 242550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−13.06689094823321, −12.82749041818396, −12.08634633497879, −11.87349615166239, −11.29578190071012, −10.94197385954682, −10.36101542399532, −9.972270990296789, −9.495309548120341, −8.697529767545524, −8.447324925006099, −8.033166849252606, −7.309452240439371, −6.795935815147810, −6.295231071738113, −6.160271427195276, −5.305409086642146, −4.949465219534574, −4.200003322638004, −4.060677975372398, −3.282541214138681, −2.852667003435557, −2.047537273063743, −1.713776164338047, −0.8690975163015989, 0,
0.8690975163015989, 1.713776164338047, 2.047537273063743, 2.852667003435557, 3.282541214138681, 4.060677975372398, 4.200003322638004, 4.949465219534574, 5.305409086642146, 6.160271427195276, 6.295231071738113, 6.795935815147810, 7.309452240439371, 8.033166849252606, 8.447324925006099, 8.697529767545524, 9.495309548120341, 9.972270990296789, 10.36101542399532, 10.94197385954682, 11.29578190071012, 11.87349615166239, 12.08634633497879, 12.82749041818396, 13.06689094823321
