| L(s) = 1 | − 5-s + 7-s − 11-s + 13-s − 5·19-s + 6·23-s + 25-s − 6·29-s + 7·31-s − 35-s + 2·37-s + 43-s + 3·47-s − 6·49-s + 12·53-s + 55-s + 9·59-s + 5·61-s − 65-s + 7·67-s − 6·71-s + 2·73-s − 77-s − 5·79-s + 9·83-s + 3·89-s + 91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 0.301·11-s + 0.277·13-s − 1.14·19-s + 1.25·23-s + 1/5·25-s − 1.11·29-s + 1.25·31-s − 0.169·35-s + 0.328·37-s + 0.152·43-s + 0.437·47-s − 6/7·49-s + 1.64·53-s + 0.134·55-s + 1.17·59-s + 0.640·61-s − 0.124·65-s + 0.855·67-s − 0.712·71-s + 0.234·73-s − 0.113·77-s − 0.562·79-s + 0.987·83-s + 0.317·89-s + 0.104·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.183626780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.183626780\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
| 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
−13.65060961526748, −13.08999148954694, −12.92560303775223, −12.24942514902733, −11.68888664730633, −11.27983844402041, −10.86237235784895, −10.36130189292213, −9.852934921577920, −9.159357887181527, −8.711029019632811, −8.254846546429713, −7.813457322388020, −7.154555961775171, −6.748404166476965, −6.164581805917989, −5.451707479483644, −5.066064840820762, −4.321880471349382, −3.987724133228370, −3.258228464114834, −2.580964108017949, −2.048234575067804, −1.158269537613944, −0.5056514991284791,
0.5056514991284791, 1.158269537613944, 2.048234575067804, 2.580964108017949, 3.258228464114834, 3.987724133228370, 4.321880471349382, 5.066064840820762, 5.451707479483644, 6.164581805917989, 6.748404166476965, 7.154555961775171, 7.813457322388020, 8.254846546429713, 8.711029019632811, 9.159357887181527, 9.852934921577920, 10.36130189292213, 10.86237235784895, 11.27983844402041, 11.68888664730633, 12.24942514902733, 12.92560303775223, 13.08999148954694, 13.65060961526748
