| L(s) = 1 | − 2-s + 1.80·3-s + 4-s + 1.03·5-s − 1.80·6-s + 3.67·7-s − 8-s + 0.265·9-s − 1.03·10-s − 1.94·11-s + 1.80·12-s + 2.38·13-s − 3.67·14-s + 1.86·15-s + 16-s − 4.68·17-s − 0.265·18-s + 8.14·19-s + 1.03·20-s + 6.64·21-s + 1.94·22-s + 4.78·23-s − 1.80·24-s − 3.93·25-s − 2.38·26-s − 4.94·27-s + 3.67·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.04·3-s + 0.5·4-s + 0.462·5-s − 0.737·6-s + 1.39·7-s − 0.353·8-s + 0.0884·9-s − 0.326·10-s − 0.587·11-s + 0.521·12-s + 0.660·13-s − 0.983·14-s + 0.482·15-s + 0.250·16-s − 1.13·17-s − 0.0625·18-s + 1.86·19-s + 0.231·20-s + 1.45·21-s + 0.415·22-s + 0.997·23-s − 0.368·24-s − 0.786·25-s − 0.467·26-s − 0.951·27-s + 0.695·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.661857969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.661857969\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2011 | \( 1 - T \) |
| good | 3 | \( 1 - 1.80T + 3T^{2} \) |
| 5 | \( 1 - 1.03T + 5T^{2} \) |
| 7 | \( 1 - 3.67T + 7T^{2} \) |
| 11 | \( 1 + 1.94T + 11T^{2} \) |
| 13 | \( 1 - 2.38T + 13T^{2} \) |
| 17 | \( 1 + 4.68T + 17T^{2} \) |
| 19 | \( 1 - 8.14T + 19T^{2} \) |
| 23 | \( 1 - 4.78T + 23T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 + 0.813T + 31T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 41 | \( 1 - 1.17T + 41T^{2} \) |
| 43 | \( 1 - 8.60T + 43T^{2} \) |
| 47 | \( 1 - 6.24T + 47T^{2} \) |
| 53 | \( 1 - 0.673T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 + 6.75T + 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 + 3.22T + 71T^{2} \) |
| 73 | \( 1 - 0.571T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 - 8.03T + 83T^{2} \) |
| 89 | \( 1 - 8.43T + 89T^{2} \) |
| 97 | \( 1 - 5.01T + 97T^{2} \) |
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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
−8.570789240253166065041861274539, −7.80362680940345325327747607158, −7.41830245904267432994902176676, −6.37503699552678657210561670767, −5.41225975399509074841427498181, −4.77963612112738250705045047129, −3.57115237412884769972280306154, −2.70017082517205593804182690244, −1.98570568207964981282703727710, −1.05091924576804330609478342983,
1.05091924576804330609478342983, 1.98570568207964981282703727710, 2.70017082517205593804182690244, 3.57115237412884769972280306154, 4.77963612112738250705045047129, 5.41225975399509074841427498181, 6.37503699552678657210561670767, 7.41830245904267432994902176676, 7.80362680940345325327747607158, 8.570789240253166065041861274539
