| L(s) = 1 | + 2.68·3-s + 5-s + 3.18·7-s + 4.18·9-s − 0.681·13-s + 2.68·15-s − 1.18·17-s − 19-s + 8.55·21-s + 2.17·23-s + 25-s + 3.18·27-s + 2.81·29-s − 6.37·31-s + 3.18·35-s + 7.87·37-s − 1.82·39-s + 0.983·41-s − 1.36·43-s + 4.18·45-s − 11.7·47-s + 3.17·49-s − 3.18·51-s − 1.69·53-s − 2.68·57-s − 11.5·59-s + 7.36·61-s + ⋯ |
| L(s) = 1 | + 1.54·3-s + 0.447·5-s + 1.20·7-s + 1.39·9-s − 0.188·13-s + 0.692·15-s − 0.288·17-s − 0.229·19-s + 1.86·21-s + 0.453·23-s + 0.200·25-s + 0.613·27-s + 0.521·29-s − 1.14·31-s + 0.539·35-s + 1.29·37-s − 0.292·39-s + 0.153·41-s − 0.207·43-s + 0.624·45-s − 1.71·47-s + 0.453·49-s − 0.446·51-s − 0.233·53-s − 0.355·57-s − 1.50·59-s + 0.942·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.538683877\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.538683877\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.68T + 3T^{2} \) |
| 7 | \( 1 - 3.18T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 0.681T + 13T^{2} \) |
| 17 | \( 1 + 1.18T + 17T^{2} \) |
| 23 | \( 1 - 2.17T + 23T^{2} \) |
| 29 | \( 1 - 2.81T + 29T^{2} \) |
| 31 | \( 1 + 6.37T + 31T^{2} \) |
| 37 | \( 1 - 7.87T + 37T^{2} \) |
| 41 | \( 1 - 0.983T + 41T^{2} \) |
| 43 | \( 1 + 1.36T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 + 1.69T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 - 7.36T + 61T^{2} \) |
| 67 | \( 1 + 7.02T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 + 5.53T + 73T^{2} \) |
| 79 | \( 1 + 5.36T + 79T^{2} \) |
| 83 | \( 1 + 2.37T + 83T^{2} \) |
| 89 | \( 1 - 3.01T + 89T^{2} \) |
| 97 | \( 1 - 4.88T + 97T^{2} \) |
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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
−9.276493317100932161319152562131, −8.662594942496993976777955171711, −7.977850034518973652271147128097, −7.39631650848874463642914827090, −6.32543172730690716256767398146, −5.10623204682097692601510458475, −4.35704139143883124490320642831, −3.26553310273189336483224958201, −2.31612204161527165908495480690, −1.51960481786494023859307073891,
1.51960481786494023859307073891, 2.31612204161527165908495480690, 3.26553310273189336483224958201, 4.35704139143883124490320642831, 5.10623204682097692601510458475, 6.32543172730690716256767398146, 7.39631650848874463642914827090, 7.977850034518973652271147128097, 8.662594942496993976777955171711, 9.276493317100932161319152562131
