| L(s) = 1 | − 2-s − 3-s + 4-s + 2.82·5-s + 6-s + 1.41·7-s − 8-s + 9-s − 2.82·10-s + 4·11-s − 12-s − 0.828·13-s − 1.41·14-s − 2.82·15-s + 16-s − 2.82·17-s − 18-s − 5.41·19-s + 2.82·20-s − 1.41·21-s − 4·22-s − 23-s + 24-s + 3.00·25-s + 0.828·26-s − 27-s + 1.41·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.26·5-s + 0.408·6-s + 0.534·7-s − 0.353·8-s + 0.333·9-s − 0.894·10-s + 1.20·11-s − 0.288·12-s − 0.229·13-s − 0.377·14-s − 0.730·15-s + 0.250·16-s − 0.685·17-s − 0.235·18-s − 1.24·19-s + 0.632·20-s − 0.308·21-s − 0.852·22-s − 0.208·23-s + 0.204·24-s + 0.600·25-s + 0.162·26-s − 0.192·27-s + 0.267·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.619961123\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.619961123\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 0.828T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 + 5.41T + 19T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 4.82T + 37T^{2} \) |
| 41 | \( 1 - 2.82T + 41T^{2} \) |
| 43 | \( 1 + 0.242T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + 1.17T + 59T^{2} \) |
| 61 | \( 1 - 3.65T + 61T^{2} \) |
| 67 | \( 1 + 2.48T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 - 15.6T + 79T^{2} \) |
| 83 | \( 1 - 17.5T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−8.595873074109167780295024981352, −7.79066281271844786795919490019, −6.72291925483292991893637793943, −6.39013183715669943158451778504, −5.73165128724882380008890325230, −4.74913849570471207850166199658, −4.00240461693196721511794656202, −2.48024738302912223821866964593, −1.85007955791145917102426300775, −0.866812144772610539924526582948,
0.866812144772610539924526582948, 1.85007955791145917102426300775, 2.48024738302912223821866964593, 4.00240461693196721511794656202, 4.74913849570471207850166199658, 5.73165128724882380008890325230, 6.39013183715669943158451778504, 6.72291925483292991893637793943, 7.79066281271844786795919490019, 8.595873074109167780295024981352
