| L(s) = 1 | + 0.0878·2-s + 3·3-s − 7.99·4-s + 16.5·5-s + 0.263·6-s + 0.843·7-s − 1.40·8-s + 9·9-s + 1.45·10-s + 15.5·11-s − 23.9·12-s + 10.3·13-s + 0.0741·14-s + 49.5·15-s + 63.8·16-s + 101.·17-s + 0.790·18-s + 22.6·19-s − 131.·20-s + 2.53·21-s + 1.36·22-s + 23·23-s − 4.21·24-s + 147.·25-s + 0.911·26-s + 27·27-s − 6.74·28-s + ⋯ |
| L(s) = 1 | + 0.0310·2-s + 0.577·3-s − 0.999·4-s + 1.47·5-s + 0.0179·6-s + 0.0455·7-s − 0.0621·8-s + 0.333·9-s + 0.0458·10-s + 0.425·11-s − 0.576·12-s + 0.221·13-s + 0.00141·14-s + 0.852·15-s + 0.997·16-s + 1.45·17-s + 0.0103·18-s + 0.273·19-s − 1.47·20-s + 0.0262·21-s + 0.0132·22-s + 0.208·23-s − 0.0358·24-s + 1.18·25-s + 0.00687·26-s + 0.192·27-s − 0.0454·28-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
| \(L(2)\) |
\(\approx\) |
\(3.595441877\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.595441877\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 0.0878T + 8T^{2} \) |
| 5 | \( 1 - 16.5T + 125T^{2} \) |
| 7 | \( 1 - 0.843T + 343T^{2} \) |
| 11 | \( 1 - 15.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 10.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 101.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 22.6T + 6.85e3T^{2} \) |
| 31 | \( 1 - 38.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 170.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 176.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 353.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 311.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 314.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 727.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 202.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 651.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 587.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 50.1T + 3.89e5T^{2} \) |
| 79 | \( 1 + 210.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 729.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 641.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 424.T + 9.12e5T^{2} \) |
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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
−8.922992917260649853498183223869, −8.224522003431532628861654859960, −7.34803219127887884795516081741, −6.24008629869892315105820902934, −5.57299296979648253220691822093, −4.85596070174522650891946777670, −3.76867896881104074487844050289, −2.95414190619135289687074411694, −1.74025534715741329941819736596, −0.921254689249900530911352339042,
0.921254689249900530911352339042, 1.74025534715741329941819736596, 2.95414190619135289687074411694, 3.76867896881104074487844050289, 4.85596070174522650891946777670, 5.57299296979648253220691822093, 6.24008629869892315105820902934, 7.34803219127887884795516081741, 8.224522003431532628861654859960, 8.922992917260649853498183223869
