| L(s) = 1 | + 3.18·5-s + 4.63·7-s + 0.696·11-s − 13-s + 6.37·17-s + 5.01·19-s + 0.730·23-s + 5.16·25-s − 0.919·29-s − 6.95·31-s + 14.7·35-s − 6.56·37-s − 11.0·41-s + 3.64·43-s + 12.6·47-s + 14.4·49-s + 2.98·53-s + 2.22·55-s − 1.76·59-s + 7.39·61-s − 3.18·65-s − 14.9·67-s − 0.353·71-s − 7.28·73-s + 3.22·77-s − 8.26·79-s − 13.7·83-s + ⋯ |
| L(s) = 1 | + 1.42·5-s + 1.75·7-s + 0.210·11-s − 0.277·13-s + 1.54·17-s + 1.15·19-s + 0.152·23-s + 1.03·25-s − 0.170·29-s − 1.24·31-s + 2.49·35-s − 1.07·37-s − 1.72·41-s + 0.556·43-s + 1.85·47-s + 2.06·49-s + 0.409·53-s + 0.299·55-s − 0.229·59-s + 0.946·61-s − 0.395·65-s − 1.82·67-s − 0.0419·71-s − 0.852·73-s + 0.367·77-s − 0.929·79-s − 1.51·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4212 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4212 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.542171835\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.542171835\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 3.18T + 5T^{2} \) |
| 7 | \( 1 - 4.63T + 7T^{2} \) |
| 11 | \( 1 - 0.696T + 11T^{2} \) |
| 17 | \( 1 - 6.37T + 17T^{2} \) |
| 19 | \( 1 - 5.01T + 19T^{2} \) |
| 23 | \( 1 - 0.730T + 23T^{2} \) |
| 29 | \( 1 + 0.919T + 29T^{2} \) |
| 31 | \( 1 + 6.95T + 31T^{2} \) |
| 37 | \( 1 + 6.56T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 - 3.64T + 43T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 - 2.98T + 53T^{2} \) |
| 59 | \( 1 + 1.76T + 59T^{2} \) |
| 61 | \( 1 - 7.39T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 + 0.353T + 71T^{2} \) |
| 73 | \( 1 + 7.28T + 73T^{2} \) |
| 79 | \( 1 + 8.26T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 - 5.80T + 89T^{2} \) |
| 97 | \( 1 + 3.03T + 97T^{2} \) |
| 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
−8.523264694836916433170639871550, −7.49606670227361202814551334059, −7.21293526901288906758431494281, −5.86716189256541314756674914550, −5.43122533616596492058148250426, −4.98439481841823682389794356626, −3.84315791774256468425346223111, −2.75723122824214002390874458275, −1.72074231355937591556154142214, −1.27298693074698225144399946877,
1.27298693074698225144399946877, 1.72074231355937591556154142214, 2.75723122824214002390874458275, 3.84315791774256468425346223111, 4.98439481841823682389794356626, 5.43122533616596492058148250426, 5.86716189256541314756674914550, 7.21293526901288906758431494281, 7.49606670227361202814551334059, 8.523264694836916433170639871550
