| L(s) = 1 | + 2-s + 1.22·3-s + 4-s − 5-s + 1.22·6-s − 7-s + 8-s − 1.50·9-s − 10-s + 1.22·12-s + 3.47·13-s − 14-s − 1.22·15-s + 16-s − 0.138·17-s − 1.50·18-s + 0.515·19-s − 20-s − 1.22·21-s + 1.51·23-s + 1.22·24-s + 25-s + 3.47·26-s − 5.50·27-s − 28-s + 9.24·29-s − 1.22·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.705·3-s + 0.5·4-s − 0.447·5-s + 0.498·6-s − 0.377·7-s + 0.353·8-s − 0.502·9-s − 0.316·10-s + 0.352·12-s + 0.964·13-s − 0.267·14-s − 0.315·15-s + 0.250·16-s − 0.0335·17-s − 0.355·18-s + 0.118·19-s − 0.223·20-s − 0.266·21-s + 0.314·23-s + 0.249·24-s + 0.200·25-s + 0.682·26-s − 1.05·27-s − 0.188·28-s + 1.71·29-s − 0.223·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.778753322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.778753322\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 1.22T + 3T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 + 0.138T + 17T^{2} \) |
| 19 | \( 1 - 0.515T + 19T^{2} \) |
| 23 | \( 1 - 1.51T + 23T^{2} \) |
| 29 | \( 1 - 9.24T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 - 1.66T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 + 1.80T + 47T^{2} \) |
| 53 | \( 1 - 1.39T + 53T^{2} \) |
| 59 | \( 1 - 9.30T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 + 1.84T + 67T^{2} \) |
| 71 | \( 1 + 6.39T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 - 9.98T + 79T^{2} \) |
| 83 | \( 1 - 2.20T + 83T^{2} \) |
| 89 | \( 1 - 1.74T + 89T^{2} \) |
| 97 | \( 1 + 4.41T + 97T^{2} \) |
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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
−7.949178428680173809592899041793, −6.84216981388300107888496945595, −6.54694871432075301679610392404, −5.69159697899526825236299030646, −4.89166169160210478180116501339, −4.18156850576733662566139553097, −3.23646563288345857764497732961, −3.08316058595281667051443601452, −2.01951979470211914309428645931, −0.823221426125558794979795274433,
0.823221426125558794979795274433, 2.01951979470211914309428645931, 3.08316058595281667051443601452, 3.23646563288345857764497732961, 4.18156850576733662566139553097, 4.89166169160210478180116501339, 5.69159697899526825236299030646, 6.54694871432075301679610392404, 6.84216981388300107888496945595, 7.949178428680173809592899041793
