| L(s) = 1 | − 2.80·2-s + 3-s + 5.84·4-s − 3.48·5-s − 2.80·6-s − 1.41·7-s − 10.7·8-s + 9-s + 9.74·10-s − 4.43·11-s + 5.84·12-s − 2.37·13-s + 3.95·14-s − 3.48·15-s + 18.4·16-s − 3.28·17-s − 2.80·18-s − 3.46·19-s − 20.3·20-s − 1.41·21-s + 12.4·22-s + 23-s − 10.7·24-s + 7.11·25-s + 6.65·26-s + 27-s − 8.26·28-s + ⋯ |
| L(s) = 1 | − 1.98·2-s + 0.577·3-s + 2.92·4-s − 1.55·5-s − 1.14·6-s − 0.534·7-s − 3.80·8-s + 0.333·9-s + 3.08·10-s − 1.33·11-s + 1.68·12-s − 0.659·13-s + 1.05·14-s − 0.898·15-s + 4.61·16-s − 0.797·17-s − 0.660·18-s − 0.794·19-s − 4.54·20-s − 0.308·21-s + 2.64·22-s + 0.208·23-s − 2.19·24-s + 1.42·25-s + 1.30·26-s + 0.192·27-s − 1.56·28-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
| \(L(1)\) |
\(\approx\) |
\(0.1911292438\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1911292438\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.80T + 2T^{2} \) |
| 5 | \( 1 + 3.48T + 5T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 + 4.43T + 11T^{2} \) |
| 13 | \( 1 + 2.37T + 13T^{2} \) |
| 17 | \( 1 + 3.28T + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 31 | \( 1 + 4.18T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 - 3.71T + 41T^{2} \) |
| 43 | \( 1 + 7.08T + 43T^{2} \) |
| 47 | \( 1 - 1.63T + 47T^{2} \) |
| 53 | \( 1 - 6.97T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 - 5.25T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 - 8.68T + 73T^{2} \) |
| 79 | \( 1 - 9.51T + 79T^{2} \) |
| 83 | \( 1 - 2.12T + 83T^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.978852077497752944693564308931, −8.287878095558728169710014981016, −7.963359545723086231424848188784, −7.06075622917191680519403798086, −6.78292685401696852719204365125, −5.27775860968162876705531763647, −3.80225660351900454616205734936, −2.90272378424153054908897067962, −2.10188158773870166787738123957, −0.35008251248634018234921751290,
0.35008251248634018234921751290, 2.10188158773870166787738123957, 2.90272378424153054908897067962, 3.80225660351900454616205734936, 5.27775860968162876705531763647, 6.78292685401696852719204365125, 7.06075622917191680519403798086, 7.963359545723086231424848188784, 8.287878095558728169710014981016, 8.978852077497752944693564308931
