| L(s) = 1 | − 0.359·2-s + 3-s − 1.87·4-s − 1.76·5-s − 0.359·6-s − 7-s + 1.39·8-s + 9-s + 0.635·10-s − 4.93·11-s − 1.87·12-s − 4.06·13-s + 0.359·14-s − 1.76·15-s + 3.23·16-s − 3.23·17-s − 0.359·18-s − 1.79·19-s + 3.30·20-s − 21-s + 1.77·22-s − 5.43·23-s + 1.39·24-s − 1.88·25-s + 1.46·26-s + 27-s + 1.87·28-s + ⋯ |
| L(s) = 1 | − 0.254·2-s + 0.577·3-s − 0.935·4-s − 0.789·5-s − 0.146·6-s − 0.377·7-s + 0.492·8-s + 0.333·9-s + 0.200·10-s − 1.48·11-s − 0.539·12-s − 1.12·13-s + 0.0962·14-s − 0.455·15-s + 0.809·16-s − 0.784·17-s − 0.0848·18-s − 0.411·19-s + 0.738·20-s − 0.218·21-s + 0.378·22-s − 1.13·23-s + 0.284·24-s − 0.376·25-s + 0.287·26-s + 0.192·27-s + 0.353·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4362158365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4362158365\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 + T \) |
| good | 2 | \( 1 + 0.359T + 2T^{2} \) |
| 5 | \( 1 + 1.76T + 5T^{2} \) |
| 11 | \( 1 + 4.93T + 11T^{2} \) |
| 13 | \( 1 + 4.06T + 13T^{2} \) |
| 17 | \( 1 + 3.23T + 17T^{2} \) |
| 19 | \( 1 + 1.79T + 19T^{2} \) |
| 23 | \( 1 + 5.43T + 23T^{2} \) |
| 29 | \( 1 - 9.41T + 29T^{2} \) |
| 31 | \( 1 + 9.32T + 31T^{2} \) |
| 37 | \( 1 - 2.56T + 37T^{2} \) |
| 41 | \( 1 - 3.91T + 41T^{2} \) |
| 43 | \( 1 - 7.54T + 43T^{2} \) |
| 47 | \( 1 - 3.41T + 47T^{2} \) |
| 53 | \( 1 + 13.6T + 53T^{2} \) |
| 59 | \( 1 + 1.37T + 59T^{2} \) |
| 61 | \( 1 + 6.83T + 61T^{2} \) |
| 67 | \( 1 - 3.86T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 3.93T + 73T^{2} \) |
| 79 | \( 1 - 3.31T + 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 + 6.36T + 97T^{2} \) |
| 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
−8.345369419796345805532279071305, −7.76992265038820180306418942400, −7.45130321642503822970484467135, −6.27933281523910201483556872168, −5.26034164012732232254663534871, −4.53448306056296228885003323840, −3.95961161326508068975689617415, −2.94220580536483383426563194626, −2.12447824746123283192188477856, −0.36142074788815506681519225886,
0.36142074788815506681519225886, 2.12447824746123283192188477856, 2.94220580536483383426563194626, 3.95961161326508068975689617415, 4.53448306056296228885003323840, 5.26034164012732232254663534871, 6.27933281523910201483556872168, 7.45130321642503822970484467135, 7.76992265038820180306418942400, 8.345369419796345805532279071305
