| L(s) = 1 | + 5.25·2-s + 3·3-s + 19.6·4-s + 20.4·5-s + 15.7·6-s − 5.25·7-s + 61.2·8-s + 9·9-s + 107.·10-s − 27.5·11-s + 58.9·12-s + 11.3·13-s − 27.6·14-s + 61.3·15-s + 165.·16-s + 83.8·17-s + 47.3·18-s − 115.·19-s + 402.·20-s − 15.7·21-s − 144.·22-s + 23·23-s + 183.·24-s + 293.·25-s + 59.5·26-s + 27·27-s − 103.·28-s + ⋯ |
| L(s) = 1 | + 1.85·2-s + 0.577·3-s + 2.45·4-s + 1.83·5-s + 1.07·6-s − 0.283·7-s + 2.70·8-s + 0.333·9-s + 3.40·10-s − 0.755·11-s + 1.41·12-s + 0.241·13-s − 0.527·14-s + 1.05·15-s + 2.57·16-s + 1.19·17-s + 0.619·18-s − 1.39·19-s + 4.49·20-s − 0.163·21-s − 1.40·22-s + 0.208·23-s + 1.56·24-s + 2.34·25-s + 0.449·26-s + 0.192·27-s − 0.697·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(13.57253565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.57253565\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 23 | \( 1 - 23T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 5.25T + 8T^{2} \) |
| 5 | \( 1 - 20.4T + 125T^{2} \) |
| 7 | \( 1 + 5.25T + 343T^{2} \) |
| 11 | \( 1 + 27.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 11.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 83.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 115.T + 6.85e3T^{2} \) |
| 31 | \( 1 - 246.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 351.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 142.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 229.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 359.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 767.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 444.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 359.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 804.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 749.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 496.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 7.17T + 4.93e5T^{2} \) |
| 83 | \( 1 + 340.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.53e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.01e3T + 9.12e5T^{2} \) |
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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
−8.765581712062547731418173694697, −7.83290978920108970410557195141, −6.54953796652443081764112931341, −6.45288716768188753107121722505, −5.33245846183058606967649484038, −5.04045373832864191318094942224, −3.78803990244987185258941501663, −2.91863269196202862166559589784, −2.28299300591720658308811821449, −1.47020249851552387755217502501,
1.47020249851552387755217502501, 2.28299300591720658308811821449, 2.91863269196202862166559589784, 3.78803990244987185258941501663, 5.04045373832864191318094942224, 5.33245846183058606967649484038, 6.45288716768188753107121722505, 6.54953796652443081764112931341, 7.83290978920108970410557195141, 8.765581712062547731418173694697
