| L(s) = 1 | + 5.38·2-s + 3·3-s + 21.0·4-s − 8.47·5-s + 16.1·6-s − 23.4·7-s + 70.2·8-s + 9·9-s − 45.6·10-s + 51.5·11-s + 63.1·12-s + 87.9·13-s − 126.·14-s − 25.4·15-s + 210.·16-s − 82.5·17-s + 48.5·18-s + 39.8·19-s − 178.·20-s − 70.4·21-s + 277.·22-s + 23·23-s + 210.·24-s − 53.2·25-s + 473.·26-s + 27·27-s − 493.·28-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 0.577·3-s + 2.63·4-s − 0.757·5-s + 1.10·6-s − 1.26·7-s + 3.10·8-s + 0.333·9-s − 1.44·10-s + 1.41·11-s + 1.51·12-s + 1.87·13-s − 2.41·14-s − 0.437·15-s + 3.28·16-s − 1.17·17-s + 0.635·18-s + 0.480·19-s − 1.99·20-s − 0.731·21-s + 2.69·22-s + 0.208·23-s + 1.79·24-s − 0.425·25-s + 3.57·26-s + 0.192·27-s − 3.33·28-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
| \(L(2)\) |
\(\approx\) |
\(9.551102383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.551102383\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5.38T + 8T^{2} \) |
| 5 | \( 1 + 8.47T + 125T^{2} \) |
| 7 | \( 1 + 23.4T + 343T^{2} \) |
| 11 | \( 1 - 51.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 87.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 82.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 39.8T + 6.85e3T^{2} \) |
| 31 | \( 1 - 37.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 316.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 409.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 457.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 432.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 70.7T + 1.48e5T^{2} \) |
| 59 | \( 1 + 809.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 582.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 50.7T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.03e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 631.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 252.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 381.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 30.3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 39.8T + 9.12e5T^{2} \) |
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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
−8.806585362766597320542544655350, −7.69542424430721155790047728911, −6.79783544236837692827994398717, −6.40837869262743237968038167014, −5.67081912945572957653407068304, −4.26807834954440526151133226500, −3.83478707075832789919784953270, −3.41600084557132981428003957136, −2.36520565036003481465866853546, −1.10437808622441574089384853759,
1.10437808622441574089384853759, 2.36520565036003481465866853546, 3.41600084557132981428003957136, 3.83478707075832789919784953270, 4.26807834954440526151133226500, 5.67081912945572957653407068304, 6.40837869262743237968038167014, 6.79783544236837692827994398717, 7.69542424430721155790047728911, 8.806585362766597320542544655350
