L(s) = 1 | − 2.62·2-s + 2.49·3-s + 4.89·4-s + 0.915·5-s − 6.55·6-s + 2.77·7-s − 7.58·8-s + 3.23·9-s − 2.40·10-s + 3.08·11-s + 12.2·12-s − 5.11·13-s − 7.27·14-s + 2.28·15-s + 10.1·16-s − 0.311·17-s − 8.49·18-s + 5.58·19-s + 4.47·20-s + 6.92·21-s − 8.10·22-s + 6.75·23-s − 18.9·24-s − 4.16·25-s + 13.4·26-s + 0.591·27-s + 13.5·28-s + ⋯ |
L(s) = 1 | − 1.85·2-s + 1.44·3-s + 2.44·4-s + 0.409·5-s − 2.67·6-s + 1.04·7-s − 2.68·8-s + 1.07·9-s − 0.760·10-s + 0.930·11-s + 3.52·12-s − 1.41·13-s − 1.94·14-s + 0.590·15-s + 2.53·16-s − 0.0756·17-s − 2.00·18-s + 1.28·19-s + 1.00·20-s + 1.51·21-s − 1.72·22-s + 1.40·23-s − 3.86·24-s − 0.832·25-s + 2.63·26-s + 0.113·27-s + 2.56·28-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
\(L(1)\) |
\(\approx\) |
\(1.849357158\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.849357158\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.62T + 2T^{2} \) |
| 3 | \( 1 - 2.49T + 3T^{2} \) |
| 5 | \( 1 - 0.915T + 5T^{2} \) |
| 7 | \( 1 - 2.77T + 7T^{2} \) |
| 11 | \( 1 - 3.08T + 11T^{2} \) |
| 13 | \( 1 + 5.11T + 13T^{2} \) |
| 17 | \( 1 + 0.311T + 17T^{2} \) |
| 19 | \( 1 - 5.58T + 19T^{2} \) |
| 23 | \( 1 - 6.75T + 23T^{2} \) |
| 29 | \( 1 + 5.41T + 29T^{2} \) |
| 31 | \( 1 - 8.83T + 31T^{2} \) |
| 41 | \( 1 - 4.63T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 - 7.62T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 - 2.95T + 59T^{2} \) |
| 61 | \( 1 + 7.40T + 61T^{2} \) |
| 67 | \( 1 + 4.29T + 67T^{2} \) |
| 71 | \( 1 + 2.69T + 71T^{2} \) |
| 73 | \( 1 - 3.23T + 73T^{2} \) |
| 79 | \( 1 + 2.50T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 + 8.44T + 89T^{2} \) |
| 97 | \( 1 - 0.983T + 97T^{2} \) |
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
−8.632336924702365688708435708089, −7.80619987456479122746580099539, −7.44060287937289131796813424996, −6.84760238141498627277480292076, −5.67611860071665909274989665948, −4.60700045712653281877037472172, −3.32428008346227181221541060097, −2.50242253381562242314964149053, −1.86482692427519699300275971055, −1.01517903036376052559564736420,
1.01517903036376052559564736420, 1.86482692427519699300275971055, 2.50242253381562242314964149053, 3.32428008346227181221541060097, 4.60700045712653281877037472172, 5.67611860071665909274989665948, 6.84760238141498627277480292076, 7.44060287937289131796813424996, 7.80619987456479122746580099539, 8.632336924702365688708435708089