| 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 + 4.82·13-s + 1.41·14-s + 2.82·15-s + 16-s + 2.82·17-s − 18-s − 2.58·19-s − 2.82·20-s + 1.41·21-s − 4·22-s − 23-s + 24-s + 3.00·25-s − 4.82·26-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 + 1.33·13-s + 0.377·14-s + 0.730·15-s + 0.250·16-s + 0.685·17-s − 0.235·18-s − 0.593·19-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 − 0.946·26-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
| \(L(1)\) |
\(\approx\) |
\(0.7870559750\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7870559750\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4.82T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 + 2.58T + 19T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 0.828T + 37T^{2} \) |
| 41 | \( 1 + 2.82T + 41T^{2} \) |
| 43 | \( 1 - 8.24T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + 6.82T + 59T^{2} \) |
| 61 | \( 1 + 7.65T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 - 6.48T + 71T^{2} \) |
| 73 | \( 1 + 2.48T + 73T^{2} \) |
| 79 | \( 1 - 4.34T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 + 5.17T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−8.295403112231557533023455056063, −7.940098076461212476335725799917, −6.85458327989795409435040342220, −6.48991672876383322995908600145, −5.74882722626502390321102652718, −4.47762634747095651978261228802, −3.80843457977853764914598732930, −3.13913407842986847171720899618, −1.54736915609214351710378580454, −0.61992474936395554395889213948,
0.61992474936395554395889213948, 1.54736915609214351710378580454, 3.13913407842986847171720899618, 3.80843457977853764914598732930, 4.47762634747095651978261228802, 5.74882722626502390321102652718, 6.48991672876383322995908600145, 6.85458327989795409435040342220, 7.940098076461212476335725799917, 8.295403112231557533023455056063
