| L(s) = 1 | − 0.563·2-s + 3-s − 1.68·4-s − 1.59·5-s − 0.563·6-s + 4.95·7-s + 2.07·8-s + 9-s + 0.897·10-s − 0.616·11-s − 1.68·12-s + 6.48·13-s − 2.78·14-s − 1.59·15-s + 2.19·16-s − 1.67·17-s − 0.563·18-s + 2.69·19-s + 2.68·20-s + 4.95·21-s + 0.347·22-s − 23-s + 2.07·24-s − 2.45·25-s − 3.65·26-s + 27-s − 8.33·28-s + ⋯ |
| L(s) = 1 | − 0.398·2-s + 0.577·3-s − 0.841·4-s − 0.713·5-s − 0.229·6-s + 1.87·7-s + 0.733·8-s + 0.333·9-s + 0.283·10-s − 0.185·11-s − 0.485·12-s + 1.79·13-s − 0.745·14-s − 0.411·15-s + 0.549·16-s − 0.406·17-s − 0.132·18-s + 0.618·19-s + 0.599·20-s + 1.08·21-s + 0.0740·22-s − 0.208·23-s + 0.423·24-s − 0.491·25-s − 0.716·26-s + 0.192·27-s − 1.57·28-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
| \(L(1)\) |
\(\approx\) |
\(1.719641703\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.719641703\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.563T + 2T^{2} \) |
| 5 | \( 1 + 1.59T + 5T^{2} \) |
| 7 | \( 1 - 4.95T + 7T^{2} \) |
| 11 | \( 1 + 0.616T + 11T^{2} \) |
| 13 | \( 1 - 6.48T + 13T^{2} \) |
| 17 | \( 1 + 1.67T + 17T^{2} \) |
| 19 | \( 1 - 2.69T + 19T^{2} \) |
| 31 | \( 1 + 3.94T + 31T^{2} \) |
| 37 | \( 1 - 5.81T + 37T^{2} \) |
| 41 | \( 1 + 6.01T + 41T^{2} \) |
| 43 | \( 1 - 0.472T + 43T^{2} \) |
| 47 | \( 1 - 4.27T + 47T^{2} \) |
| 53 | \( 1 + 7.82T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 6.36T + 67T^{2} \) |
| 71 | \( 1 - 8.81T + 71T^{2} \) |
| 73 | \( 1 + 2.13T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 + 6.30T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−8.784531035225727026853689729274, −8.436774301529022308606587208672, −7.897034179379924863861500069493, −7.27797107935948935656343625476, −5.81910646131721840105296338905, −4.94742737773135105923811247683, −4.15043459765277338435314685982, −3.58369065475581840743034221158, −1.92758808128471459436363400176, −0.997136888898483409135568072307,
0.997136888898483409135568072307, 1.92758808128471459436363400176, 3.58369065475581840743034221158, 4.15043459765277338435314685982, 4.94742737773135105923811247683, 5.81910646131721840105296338905, 7.27797107935948935656343625476, 7.897034179379924863861500069493, 8.436774301529022308606587208672, 8.784531035225727026853689729274
