| L(s) = 1 | − 2.35·2-s − 3-s + 3.54·4-s − 1.85·5-s + 2.35·6-s + 7-s − 3.63·8-s + 9-s + 4.36·10-s − 4.89·11-s − 3.54·12-s + 4.60·13-s − 2.35·14-s + 1.85·15-s + 1.46·16-s − 1.22·17-s − 2.35·18-s − 2.61·19-s − 6.56·20-s − 21-s + 11.5·22-s − 1.19·23-s + 3.63·24-s − 1.56·25-s − 10.8·26-s − 27-s + 3.54·28-s + ⋯ |
| L(s) = 1 | − 1.66·2-s − 0.577·3-s + 1.77·4-s − 0.828·5-s + 0.961·6-s + 0.377·7-s − 1.28·8-s + 0.333·9-s + 1.37·10-s − 1.47·11-s − 1.02·12-s + 1.27·13-s − 0.629·14-s + 0.478·15-s + 0.365·16-s − 0.296·17-s − 0.554·18-s − 0.599·19-s − 1.46·20-s − 0.218·21-s + 2.45·22-s − 0.248·23-s + 0.740·24-s − 0.312·25-s − 2.12·26-s − 0.192·27-s + 0.669·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3168734008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3168734008\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 2.35T + 2T^{2} \) |
| 5 | \( 1 + 1.85T + 5T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 - 4.60T + 13T^{2} \) |
| 17 | \( 1 + 1.22T + 17T^{2} \) |
| 19 | \( 1 + 2.61T + 19T^{2} \) |
| 23 | \( 1 + 1.19T + 23T^{2} \) |
| 29 | \( 1 - 3.14T + 29T^{2} \) |
| 31 | \( 1 - 4.94T + 31T^{2} \) |
| 37 | \( 1 - 4.92T + 37T^{2} \) |
| 41 | \( 1 - 7.78T + 41T^{2} \) |
| 43 | \( 1 + 9.42T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 8.79T + 59T^{2} \) |
| 61 | \( 1 + 6.61T + 61T^{2} \) |
| 67 | \( 1 - 8.06T + 67T^{2} \) |
| 71 | \( 1 + 4.14T + 71T^{2} \) |
| 73 | \( 1 - 7.78T + 73T^{2} \) |
| 79 | \( 1 - 1.22T + 79T^{2} \) |
| 83 | \( 1 - 7.07T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
| 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
−7.938924367003018216382043354619, −7.59332928655488034358467347795, −6.45040994094197901616792090898, −6.22787422771297423051862484562, −5.01510631954700733811579155128, −4.39524819710944391412500651274, −3.31251026780646079177812289726, −2.33127391864841216504984214696, −1.39059624540525826002521170303, −0.39283544812591830362345564407,
0.39283544812591830362345564407, 1.39059624540525826002521170303, 2.33127391864841216504984214696, 3.31251026780646079177812289726, 4.39524819710944391412500651274, 5.01510631954700733811579155128, 6.22787422771297423051862484562, 6.45040994094197901616792090898, 7.59332928655488034358467347795, 7.938924367003018216382043354619
