| L(s) = 1 | + 2-s + 0.825·3-s + 4-s + 5-s + 0.825·6-s − 3.88·7-s + 8-s − 2.31·9-s + 10-s + 0.607·11-s + 0.825·12-s + 4.39·13-s − 3.88·14-s + 0.825·15-s + 16-s + 2.75·17-s − 2.31·18-s + 1.37·19-s + 20-s − 3.20·21-s + 0.607·22-s − 2.37·23-s + 0.825·24-s + 25-s + 4.39·26-s − 4.39·27-s − 3.88·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.476·3-s + 0.5·4-s + 0.447·5-s + 0.337·6-s − 1.46·7-s + 0.353·8-s − 0.772·9-s + 0.316·10-s + 0.183·11-s + 0.238·12-s + 1.21·13-s − 1.03·14-s + 0.213·15-s + 0.250·16-s + 0.668·17-s − 0.546·18-s + 0.315·19-s + 0.223·20-s − 0.699·21-s + 0.129·22-s − 0.496·23-s + 0.168·24-s + 0.200·25-s + 0.862·26-s − 0.845·27-s − 0.733·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.431797585\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.431797585\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 - 0.825T + 3T^{2} \) |
| 7 | \( 1 + 3.88T + 7T^{2} \) |
| 11 | \( 1 - 0.607T + 11T^{2} \) |
| 13 | \( 1 - 4.39T + 13T^{2} \) |
| 17 | \( 1 - 2.75T + 17T^{2} \) |
| 19 | \( 1 - 1.37T + 19T^{2} \) |
| 23 | \( 1 + 2.37T + 23T^{2} \) |
| 29 | \( 1 - 8.10T + 29T^{2} \) |
| 31 | \( 1 - 3.18T + 31T^{2} \) |
| 37 | \( 1 - 7.52T + 37T^{2} \) |
| 41 | \( 1 - 7.59T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 - 1.07T + 47T^{2} \) |
| 53 | \( 1 + 2.26T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 - 3.79T + 61T^{2} \) |
| 67 | \( 1 + 9.28T + 67T^{2} \) |
| 71 | \( 1 + 6.94T + 71T^{2} \) |
| 73 | \( 1 - 0.125T + 73T^{2} \) |
| 79 | \( 1 + 7.02T + 79T^{2} \) |
| 83 | \( 1 + 1.08T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−8.506720337409283420023862341429, −7.70131178082656955924779445553, −6.68646647221477486626730084740, −6.09726271318411085187767078506, −5.76605097403196239275302915898, −4.59102895354271518324846873975, −3.56393713237907320589501469931, −3.14297806175120721711716376376, −2.35318623265805582082498679411, −0.943931281731908193686421645311,
0.943931281731908193686421645311, 2.35318623265805582082498679411, 3.14297806175120721711716376376, 3.56393713237907320589501469931, 4.59102895354271518324846873975, 5.76605097403196239275302915898, 6.09726271318411085187767078506, 6.68646647221477486626730084740, 7.70131178082656955924779445553, 8.506720337409283420023862341429
