| L(s) = 1 | − 2-s + 3-s + 4-s + 1.51·5-s − 6-s − 1.88·7-s − 8-s + 9-s − 1.51·10-s − 3.63·11-s + 12-s + 2.59·13-s + 1.88·14-s + 1.51·15-s + 16-s + 1.88·17-s − 18-s − 5.07·19-s + 1.51·20-s − 1.88·21-s + 3.63·22-s + 23-s − 24-s − 2.70·25-s − 2.59·26-s + 27-s − 1.88·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.677·5-s − 0.408·6-s − 0.710·7-s − 0.353·8-s + 0.333·9-s − 0.479·10-s − 1.09·11-s + 0.288·12-s + 0.720·13-s + 0.502·14-s + 0.391·15-s + 0.250·16-s + 0.456·17-s − 0.235·18-s − 1.16·19-s + 0.338·20-s − 0.410·21-s + 0.774·22-s + 0.208·23-s − 0.204·24-s − 0.540·25-s − 0.509·26-s + 0.192·27-s − 0.355·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.51T + 5T^{2} \) |
| 7 | \( 1 + 1.88T + 7T^{2} \) |
| 11 | \( 1 + 3.63T + 11T^{2} \) |
| 13 | \( 1 - 2.59T + 13T^{2} \) |
| 17 | \( 1 - 1.88T + 17T^{2} \) |
| 19 | \( 1 + 5.07T + 19T^{2} \) |
| 31 | \( 1 + 5.32T + 31T^{2} \) |
| 37 | \( 1 - 5.01T + 37T^{2} \) |
| 41 | \( 1 + 8.89T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 3.80T + 47T^{2} \) |
| 53 | \( 1 + 0.495T + 53T^{2} \) |
| 59 | \( 1 + 7.86T + 59T^{2} \) |
| 61 | \( 1 + 2.36T + 61T^{2} \) |
| 67 | \( 1 + 7.89T + 67T^{2} \) |
| 71 | \( 1 - 9.92T + 71T^{2} \) |
| 73 | \( 1 - 2.18T + 73T^{2} \) |
| 79 | \( 1 + 1.58T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 + 1.30T + 89T^{2} \) |
| 97 | \( 1 - 2.69T + 97T^{2} \) |
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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
−8.099795261076238171843199383515, −7.59315874860974024537406451946, −6.62868832171871366596356275213, −6.04416355858884545527604765963, −5.28141969200408988435304828339, −4.07031819647016386005167982061, −3.15381545662775153655981181975, −2.39940790281596533462570531884, −1.51882178662764240832146626737, 0,
1.51882178662764240832146626737, 2.39940790281596533462570531884, 3.15381545662775153655981181975, 4.07031819647016386005167982061, 5.28141969200408988435304828339, 6.04416355858884545527604765963, 6.62868832171871366596356275213, 7.59315874860974024537406451946, 8.099795261076238171843199383515
