| L(s) = 1 | − 7-s + 2·11-s − 2·13-s + 6·17-s + 4·19-s − 6·23-s + 4·31-s − 10·37-s − 2·41-s − 4·43-s − 4·47-s + 49-s − 12·53-s + 12·59-s + 6·61-s − 4·67-s − 14·71-s + 2·73-s − 2·77-s + 8·79-s + 16·83-s + 6·89-s + 2·91-s + 18·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.603·11-s − 0.554·13-s + 1.45·17-s + 0.917·19-s − 1.25·23-s + 0.718·31-s − 1.64·37-s − 0.312·41-s − 0.609·43-s − 0.583·47-s + 1/7·49-s − 1.64·53-s + 1.56·59-s + 0.768·61-s − 0.488·67-s − 1.66·71-s + 0.234·73-s − 0.227·77-s + 0.900·79-s + 1.75·83-s + 0.635·89-s + 0.209·91-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−15.81571432843850, −14.90071183574192, −14.62948898016045, −13.94028490295180, −13.68582337572109, −12.88126133399298, −12.25008961125064, −11.89344929386987, −11.56885878766867, −10.56892120159714, −10.02300341811381, −9.798231138494995, −9.097567928049607, −8.432177118701034, −7.773168495064419, −7.383194144509951, −6.534220160670071, −6.183188309160443, −5.240800036957534, −5.002405684808930, −3.878214777516205, −3.502908072963281, −2.781512575698314, −1.835259186966716, −1.095815766705821, 0,
1.095815766705821, 1.835259186966716, 2.781512575698314, 3.502908072963281, 3.878214777516205, 5.002405684808930, 5.240800036957534, 6.183188309160443, 6.534220160670071, 7.383194144509951, 7.773168495064419, 8.432177118701034, 9.097567928049607, 9.798231138494995, 10.02300341811381, 10.56892120159714, 11.56885878766867, 11.89344929386987, 12.25008961125064, 12.88126133399298, 13.68582337572109, 13.94028490295180, 14.62948898016045, 14.90071183574192, 15.81571432843850
