| L(s) = 1 | + 6.87e10·2-s − 2.46e16·3-s + 4.72e21·4-s − 4.21e24·5-s − 1.69e27·6-s − 1.19e31·7-s + 3.24e32·8-s − 6.69e34·9-s − 2.89e35·10-s − 1.68e38·11-s − 1.16e38·12-s + 6.97e40·13-s − 8.19e41·14-s + 1.03e41·15-s + 2.23e43·16-s + 7.07e44·17-s − 4.60e45·18-s + 5.30e46·19-s − 1.99e46·20-s + 2.93e47·21-s − 1.15e49·22-s + 3.48e49·23-s − 8.00e48·24-s − 1.04e51·25-s + 4.79e51·26-s + 3.31e51·27-s − 5.62e52·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0948·3-s + 0.5·4-s − 0.129·5-s − 0.0670·6-s − 1.69·7-s + 0.353·8-s − 0.991·9-s − 0.0915·10-s − 1.64·11-s − 0.0474·12-s + 1.52·13-s − 1.20·14-s + 0.0122·15-s + 0.250·16-s + 0.867·17-s − 0.700·18-s + 1.12·19-s − 0.0647·20-s + 0.161·21-s − 1.16·22-s + 0.689·23-s − 0.0335·24-s − 0.983·25-s + 1.08·26-s + 0.188·27-s − 0.849·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(74-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+73/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(37)\) |
\(\approx\) |
\(2.058699919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.058699919\) |
| \(L(\frac{75}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 6.87e10T \) |
| good | 3 | \( 1 + 2.46e16T + 6.75e34T^{2} \) |
| 5 | \( 1 + 4.21e24T + 1.05e51T^{2} \) |
| 7 | \( 1 + 1.19e31T + 4.92e61T^{2} \) |
| 11 | \( 1 + 1.68e38T + 1.05e76T^{2} \) |
| 13 | \( 1 - 6.97e40T + 2.07e81T^{2} \) |
| 17 | \( 1 - 7.07e44T + 6.64e89T^{2} \) |
| 19 | \( 1 - 5.30e46T + 2.23e93T^{2} \) |
| 23 | \( 1 - 3.48e49T + 2.54e99T^{2} \) |
| 29 | \( 1 - 8.51e52T + 5.68e106T^{2} \) |
| 31 | \( 1 - 1.30e54T + 7.40e108T^{2} \) |
| 37 | \( 1 - 2.88e57T + 3.01e114T^{2} \) |
| 41 | \( 1 - 2.55e58T + 5.41e117T^{2} \) |
| 43 | \( 1 + 1.38e59T + 1.75e119T^{2} \) |
| 47 | \( 1 + 2.03e61T + 1.15e122T^{2} \) |
| 53 | \( 1 - 8.52e61T + 7.44e125T^{2} \) |
| 59 | \( 1 + 2.99e64T + 1.87e129T^{2} \) |
| 61 | \( 1 - 3.92e64T + 2.13e130T^{2} \) |
| 67 | \( 1 - 4.85e66T + 2.01e133T^{2} \) |
| 71 | \( 1 - 4.78e67T + 1.38e135T^{2} \) |
| 73 | \( 1 - 1.55e68T + 1.05e136T^{2} \) |
| 79 | \( 1 + 1.59e69T + 3.36e138T^{2} \) |
| 83 | \( 1 + 7.89e69T + 1.23e140T^{2} \) |
| 89 | \( 1 - 4.19e69T + 2.02e142T^{2} \) |
| 97 | \( 1 + 1.72e72T + 1.08e145T^{2} \) |
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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
−13.70823007281549113256096612803, −12.82803843444250678438423077853, −11.27464587003865126597051215277, −9.828484552037627156818470253849, −7.993414950234730230019865689631, −6.30504111619821142827363168521, −5.41818531829205162311391076821, −3.44695335417012824069249004137, −2.81366920948306755492264777812, −0.65931688450511965720780678326,
0.65931688450511965720780678326, 2.81366920948306755492264777812, 3.44695335417012824069249004137, 5.41818531829205162311391076821, 6.30504111619821142827363168521, 7.993414950234730230019865689631, 9.828484552037627156818470253849, 11.27464587003865126597051215277, 12.82803843444250678438423077853, 13.70823007281549113256096612803
