L(s) = 1 | − 1.67·2-s + 1.31·3-s + 0.808·4-s − 4.46·5-s − 2.21·6-s + 0.153·7-s + 1.99·8-s − 1.26·9-s + 7.47·10-s − 2.02·11-s + 1.06·12-s − 0.641·13-s − 0.257·14-s − 5.88·15-s − 4.96·16-s + 4.82·17-s + 2.11·18-s − 5.72·19-s − 3.60·20-s + 0.202·21-s + 3.39·22-s − 7.85·23-s + 2.63·24-s + 14.9·25-s + 1.07·26-s − 5.61·27-s + 0.124·28-s + ⋯ |
L(s) = 1 | − 1.18·2-s + 0.761·3-s + 0.404·4-s − 1.99·5-s − 0.902·6-s + 0.0580·7-s + 0.706·8-s − 0.420·9-s + 2.36·10-s − 0.611·11-s + 0.307·12-s − 0.177·13-s − 0.0687·14-s − 1.51·15-s − 1.24·16-s + 1.17·17-s + 0.497·18-s − 1.31·19-s − 0.806·20-s + 0.0441·21-s + 0.724·22-s − 1.63·23-s + 0.537·24-s + 2.98·25-s + 0.210·26-s − 1.08·27-s + 0.0234·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2165291066\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2165291066\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 - T \) |
| 109 | \( 1 + T \) |
good | 2 | \( 1 + 1.67T + 2T^{2} \) |
| 3 | \( 1 - 1.31T + 3T^{2} \) |
| 5 | \( 1 + 4.46T + 5T^{2} \) |
| 7 | \( 1 - 0.153T + 7T^{2} \) |
| 11 | \( 1 + 2.02T + 11T^{2} \) |
| 13 | \( 1 + 0.641T + 13T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 + 5.72T + 19T^{2} \) |
| 23 | \( 1 + 7.85T + 23T^{2} \) |
| 29 | \( 1 - 3.40T + 29T^{2} \) |
| 31 | \( 1 + 9.46T + 31T^{2} \) |
| 41 | \( 1 + 6.28T + 41T^{2} \) |
| 43 | \( 1 + 4.53T + 43T^{2} \) |
| 47 | \( 1 - 6.13T + 47T^{2} \) |
| 53 | \( 1 - 1.29T + 53T^{2} \) |
| 59 | \( 1 + 2.52T + 59T^{2} \) |
| 61 | \( 1 + 7.78T + 61T^{2} \) |
| 67 | \( 1 + 5.17T + 67T^{2} \) |
| 71 | \( 1 - 6.99T + 71T^{2} \) |
| 73 | \( 1 + 3.90T + 73T^{2} \) |
| 79 | \( 1 - 2.27T + 79T^{2} \) |
| 83 | \( 1 - 1.48T + 83T^{2} \) |
| 89 | \( 1 - 16.3T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−8.217816494576111066104147669600, −8.020950952755982911259159658275, −7.53967837377789433308136340102, −6.68477156096073134317044365956, −5.38723454179377721528972970503, −4.39017594065034101902707335300, −3.76439396444883834145024059917, −2.96057666875337755130776123012, −1.81304480296876577602039263626, −0.29668226578293282696845750006,
0.29668226578293282696845750006, 1.81304480296876577602039263626, 2.96057666875337755130776123012, 3.76439396444883834145024059917, 4.39017594065034101902707335300, 5.38723454179377721528972970503, 6.68477156096073134317044365956, 7.53967837377789433308136340102, 8.020950952755982911259159658275, 8.217816494576111066104147669600