| L(s) = 1 | − 2-s − 3-s + 4-s + 0.230·5-s + 6-s + 4.59·7-s − 8-s + 9-s − 0.230·10-s − 4.50·11-s − 12-s + 1.15·13-s − 4.59·14-s − 0.230·15-s + 16-s − 7.42·17-s − 18-s − 5.94·19-s + 0.230·20-s − 4.59·21-s + 4.50·22-s + 23-s + 24-s − 4.94·25-s − 1.15·26-s − 27-s + 4.59·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.102·5-s + 0.408·6-s + 1.73·7-s − 0.353·8-s + 0.333·9-s − 0.0728·10-s − 1.35·11-s − 0.288·12-s + 0.320·13-s − 1.22·14-s − 0.0594·15-s + 0.250·16-s − 1.80·17-s − 0.235·18-s − 1.36·19-s + 0.0514·20-s − 1.00·21-s + 0.960·22-s + 0.208·23-s + 0.204·24-s − 0.989·25-s − 0.226·26-s − 0.192·27-s + 0.868·28-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
| \(L(1)\) |
\(\approx\) |
\(1.048099366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.048099366\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.230T + 5T^{2} \) |
| 7 | \( 1 - 4.59T + 7T^{2} \) |
| 11 | \( 1 + 4.50T + 11T^{2} \) |
| 13 | \( 1 - 1.15T + 13T^{2} \) |
| 17 | \( 1 + 7.42T + 17T^{2} \) |
| 19 | \( 1 + 5.94T + 19T^{2} \) |
| 31 | \( 1 - 6.50T + 31T^{2} \) |
| 37 | \( 1 - 7.29T + 37T^{2} \) |
| 41 | \( 1 - 0.0953T + 41T^{2} \) |
| 43 | \( 1 - 0.941T + 43T^{2} \) |
| 47 | \( 1 + 0.598T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 + 2.94T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 - 2.09T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 16.5T + 79T^{2} \) |
| 83 | \( 1 + 1.71T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 + 1.82T + 97T^{2} \) |
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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
−8.308141225689569048871619558886, −7.958710481808238266256809122228, −7.09062906218052413076981248971, −6.28636572405820789937375183650, −5.50863859880605912001720531561, −4.70674940525600118228113621584, −4.14820613937935349855855121250, −2.38882215644980644418351265159, −2.01502943840024056540401026618, −0.65810877460570271866895200525,
0.65810877460570271866895200525, 2.01502943840024056540401026618, 2.38882215644980644418351265159, 4.14820613937935349855855121250, 4.70674940525600118228113621584, 5.50863859880605912001720531561, 6.28636572405820789937375183650, 7.09062906218052413076981248971, 7.958710481808238266256809122228, 8.308141225689569048871619558886
