| L(s) = 1 | − 3·9-s − 4·11-s − 6·13-s + 6·17-s − 19-s + 8·23-s + 2·29-s + 2·37-s + 2·41-s − 4·43-s − 8·47-s − 7·49-s − 6·53-s − 4·59-s + 2·61-s − 8·67-s − 8·71-s − 2·73-s + 8·79-s + 9·81-s − 4·83-s − 14·89-s − 14·97-s + 12·99-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 9-s − 1.20·11-s − 1.66·13-s + 1.45·17-s − 0.229·19-s + 1.66·23-s + 0.371·29-s + 0.328·37-s + 0.312·41-s − 0.609·43-s − 1.16·47-s − 49-s − 0.824·53-s − 0.520·59-s + 0.256·61-s − 0.977·67-s − 0.949·71-s − 0.234·73-s + 0.900·79-s + 81-s − 0.439·83-s − 1.48·89-s − 1.42·97-s + 1.20·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9000074648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9000074648\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 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 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.96881342674979, −14.58041058863147, −14.30352419800205, −13.43118083865814, −13.04012625429066, −12.33438802092835, −12.13454419855153, −11.23657846181486, −10.99516937710589, −10.12689333698412, −9.860518371495269, −9.233240889401617, −8.478640878532014, −8.005818107770003, −7.481110021161676, −6.984098751604547, −6.116013124631831, −5.549177147883454, −4.872583811238682, −4.758641066737797, −3.377000382075955, −2.950740034536898, −2.496567915155094, −1.458306377881768, −0.3574800101125213,
0.3574800101125213, 1.458306377881768, 2.496567915155094, 2.950740034536898, 3.377000382075955, 4.758641066737797, 4.872583811238682, 5.549177147883454, 6.116013124631831, 6.984098751604547, 7.481110021161676, 8.005818107770003, 8.478640878532014, 9.233240889401617, 9.860518371495269, 10.12689333698412, 10.99516937710589, 11.23657846181486, 12.13454419855153, 12.33438802092835, 13.04012625429066, 13.43118083865814, 14.30352419800205, 14.58041058863147, 14.96881342674979
