| L(s) = 1 | + 15.8·5-s + 7·7-s + 15.8·11-s + 26·13-s − 79.3·17-s + 68·19-s + 47.6·23-s + 127.·25-s − 253.·29-s + 212·31-s + 111.·35-s + 218·37-s + 396.·41-s + 260·43-s + 412.·47-s + 49·49-s − 476.·53-s + 252.·55-s − 285.·59-s − 322·61-s + 412.·65-s + 356·67-s − 1.12e3·71-s − 226·73-s + 111.·77-s + 440·79-s − 253.·83-s + ⋯ |
| L(s) = 1 | + 1.41·5-s + 0.377·7-s + 0.435·11-s + 0.554·13-s − 1.13·17-s + 0.821·19-s + 0.431·23-s + 1.01·25-s − 1.62·29-s + 1.22·31-s + 0.536·35-s + 0.968·37-s + 1.51·41-s + 0.922·43-s + 1.28·47-s + 0.142·49-s − 1.23·53-s + 0.617·55-s − 0.630·59-s − 0.675·61-s + 0.787·65-s + 0.649·67-s − 1.88·71-s − 0.362·73-s + 0.164·77-s + 0.626·79-s − 0.335·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.495586981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.495586981\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 5 | \( 1 - 15.8T + 125T^{2} \) |
| 11 | \( 1 - 15.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 26T + 2.19e3T^{2} \) |
| 17 | \( 1 + 79.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 68T + 6.85e3T^{2} \) |
| 23 | \( 1 - 47.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 253.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 212T + 2.97e4T^{2} \) |
| 37 | \( 1 - 218T + 5.06e4T^{2} \) |
| 41 | \( 1 - 396.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 260T + 7.95e4T^{2} \) |
| 47 | \( 1 - 412.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 476.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 285.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 322T + 2.26e5T^{2} \) |
| 67 | \( 1 - 356T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.12e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 226T + 3.89e5T^{2} \) |
| 79 | \( 1 - 440T + 4.93e5T^{2} \) |
| 83 | \( 1 + 253.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 206.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.33e3T + 9.12e5T^{2} \) |
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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
−11.42086147646382569505623768644, −10.69659792131246063523854487557, −9.489088443346680409681878927088, −9.021465807306124843412947455783, −7.61555064592006891886496693167, −6.36863771726325589793122002884, −5.61362793115159714422014026867, −4.31248657651245955722318739806, −2.58146475321503156828531832470, −1.31686503132134081318149149323,
1.31686503132134081318149149323, 2.58146475321503156828531832470, 4.31248657651245955722318739806, 5.61362793115159714422014026867, 6.36863771726325589793122002884, 7.61555064592006891886496693167, 9.021465807306124843412947455783, 9.489088443346680409681878927088, 10.69659792131246063523854487557, 11.42086147646382569505623768644
