| L(s) = 1 | − 0.661·2-s − 3-s − 1.56·4-s − 1.16·5-s + 0.661·6-s + 4.79·7-s + 2.35·8-s + 9-s + 0.768·10-s − 0.481·11-s + 1.56·12-s − 3.88·13-s − 3.17·14-s + 1.16·15-s + 1.56·16-s + 7.41·17-s − 0.661·18-s + 4.27·19-s + 1.81·20-s − 4.79·21-s + 0.318·22-s + 23-s − 2.35·24-s − 3.65·25-s + 2.57·26-s − 27-s − 7.49·28-s + ⋯ |
| L(s) = 1 | − 0.468·2-s − 0.577·3-s − 0.780·4-s − 0.519·5-s + 0.270·6-s + 1.81·7-s + 0.833·8-s + 0.333·9-s + 0.243·10-s − 0.145·11-s + 0.450·12-s − 1.07·13-s − 0.849·14-s + 0.299·15-s + 0.390·16-s + 1.79·17-s − 0.156·18-s + 0.979·19-s + 0.405·20-s − 1.04·21-s + 0.0679·22-s + 0.208·23-s − 0.481·24-s − 0.730·25-s + 0.504·26-s − 0.192·27-s − 1.41·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9970467399\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9970467399\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 0.661T + 2T^{2} \) |
| 5 | \( 1 + 1.16T + 5T^{2} \) |
| 7 | \( 1 - 4.79T + 7T^{2} \) |
| 11 | \( 1 + 0.481T + 11T^{2} \) |
| 13 | \( 1 + 3.88T + 13T^{2} \) |
| 17 | \( 1 - 7.41T + 17T^{2} \) |
| 19 | \( 1 - 4.27T + 19T^{2} \) |
| 31 | \( 1 + 0.692T + 31T^{2} \) |
| 37 | \( 1 + 5.71T + 37T^{2} \) |
| 41 | \( 1 - 6.42T + 41T^{2} \) |
| 43 | \( 1 + 8.76T + 43T^{2} \) |
| 47 | \( 1 - 0.616T + 47T^{2} \) |
| 53 | \( 1 - 6.31T + 53T^{2} \) |
| 59 | \( 1 - 1.01T + 59T^{2} \) |
| 61 | \( 1 + 6.33T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 - 6.45T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 + 2.44T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 - 1.71T + 89T^{2} \) |
| 97 | \( 1 - 3.96T + 97T^{2} \) |
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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
−9.170393484403170376804146988585, −8.162167781566003162530764742117, −7.71415945642561547789839299111, −7.28486354471635431228541571790, −5.57011778450257411094979589947, −5.15221390108195133445830431398, −4.49203732300997790852278073299, −3.48050575078421359514014117924, −1.79424540100270947799411083247, −0.78632197720961587697795483875,
0.78632197720961587697795483875, 1.79424540100270947799411083247, 3.48050575078421359514014117924, 4.49203732300997790852278073299, 5.15221390108195133445830431398, 5.57011778450257411094979589947, 7.28486354471635431228541571790, 7.71415945642561547789839299111, 8.162167781566003162530764742117, 9.170393484403170376804146988585
