| L(s) = 1 | − 2.42·2-s − 1.95·3-s + 3.88·4-s + 1.35·5-s + 4.74·6-s + 4.91·7-s − 4.56·8-s + 0.833·9-s − 3.28·10-s − 3.37·11-s − 7.59·12-s + 5.27·13-s − 11.9·14-s − 2.65·15-s + 3.30·16-s + 0.0667·17-s − 2.02·18-s + 1.69·19-s + 5.26·20-s − 9.62·21-s + 8.18·22-s − 2.77·23-s + 8.93·24-s − 3.16·25-s − 12.8·26-s + 4.24·27-s + 19.0·28-s + ⋯ |
| L(s) = 1 | − 1.71·2-s − 1.13·3-s + 1.94·4-s + 0.606·5-s + 1.93·6-s + 1.85·7-s − 1.61·8-s + 0.277·9-s − 1.03·10-s − 1.01·11-s − 2.19·12-s + 1.46·13-s − 3.18·14-s − 0.685·15-s + 0.825·16-s + 0.0161·17-s − 0.476·18-s + 0.388·19-s + 1.17·20-s − 2.09·21-s + 1.74·22-s − 0.578·23-s + 1.82·24-s − 0.632·25-s − 2.51·26-s + 0.816·27-s + 3.60·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6788578286\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6788578286\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 4001 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.42T + 2T^{2} \) |
| 3 | \( 1 + 1.95T + 3T^{2} \) |
| 5 | \( 1 - 1.35T + 5T^{2} \) |
| 7 | \( 1 - 4.91T + 7T^{2} \) |
| 11 | \( 1 + 3.37T + 11T^{2} \) |
| 13 | \( 1 - 5.27T + 13T^{2} \) |
| 17 | \( 1 - 0.0667T + 17T^{2} \) |
| 19 | \( 1 - 1.69T + 19T^{2} \) |
| 23 | \( 1 + 2.77T + 23T^{2} \) |
| 29 | \( 1 + 7.59T + 29T^{2} \) |
| 31 | \( 1 + 7.71T + 31T^{2} \) |
| 37 | \( 1 + 1.61T + 37T^{2} \) |
| 41 | \( 1 + 5.37T + 41T^{2} \) |
| 43 | \( 1 - 4.89T + 43T^{2} \) |
| 47 | \( 1 - 1.94T + 47T^{2} \) |
| 53 | \( 1 - 3.19T + 53T^{2} \) |
| 59 | \( 1 - 5.14T + 59T^{2} \) |
| 61 | \( 1 - 2.84T + 61T^{2} \) |
| 67 | \( 1 - 7.90T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 - 8.94T + 73T^{2} \) |
| 79 | \( 1 - 0.733T + 79T^{2} \) |
| 83 | \( 1 + 5.25T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 - 8.48T + 97T^{2} \) |
| 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
−8.436447946582502659253328236114, −7.87141662953943061260780452571, −7.28303186342334494818299698671, −6.28434139145029438051672056093, −5.51958118862681646348828240589, −5.20133563616130771360147762313, −3.85057210071622056392506213275, −2.19978711449827680327054320863, −1.65889071042767267983164768489, −0.67351224623173900268712525056,
0.67351224623173900268712525056, 1.65889071042767267983164768489, 2.19978711449827680327054320863, 3.85057210071622056392506213275, 5.20133563616130771360147762313, 5.51958118862681646348828240589, 6.28434139145029438051672056093, 7.28303186342334494818299698671, 7.87141662953943061260780452571, 8.436447946582502659253328236114
