L(s) = 1 | − 2-s − 3-s + 4-s + 0.580·5-s + 6-s + 4.77·7-s − 8-s + 9-s − 0.580·10-s + 1.62·11-s − 12-s + 5.47·13-s − 4.77·14-s − 0.580·15-s + 16-s + 17-s − 18-s + 1.16·19-s + 0.580·20-s − 4.77·21-s − 1.62·22-s − 7.82·23-s + 24-s − 4.66·25-s − 5.47·26-s − 27-s + 4.77·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.259·5-s + 0.408·6-s + 1.80·7-s − 0.353·8-s + 0.333·9-s − 0.183·10-s + 0.491·11-s − 0.288·12-s + 1.51·13-s − 1.27·14-s − 0.149·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s + 0.267·19-s + 0.129·20-s − 1.04·21-s − 0.347·22-s − 1.63·23-s + 0.204·24-s − 0.932·25-s − 1.07·26-s − 0.192·27-s + 0.903·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.892114627\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.892114627\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 5 | \( 1 - 0.580T + 5T^{2} \) |
| 7 | \( 1 - 4.77T + 7T^{2} \) |
| 11 | \( 1 - 1.62T + 11T^{2} \) |
| 13 | \( 1 - 5.47T + 13T^{2} \) |
| 19 | \( 1 - 1.16T + 19T^{2} \) |
| 23 | \( 1 + 7.82T + 23T^{2} \) |
| 29 | \( 1 - 1.14T + 29T^{2} \) |
| 31 | \( 1 - 7.10T + 31T^{2} \) |
| 37 | \( 1 - 1.66T + 37T^{2} \) |
| 41 | \( 1 + 0.211T + 41T^{2} \) |
| 43 | \( 1 - 9.35T + 43T^{2} \) |
| 47 | \( 1 - 4.12T + 47T^{2} \) |
| 53 | \( 1 - 6.63T + 53T^{2} \) |
| 61 | \( 1 - 1.75T + 61T^{2} \) |
| 67 | \( 1 - 0.972T + 67T^{2} \) |
| 71 | \( 1 - 0.225T + 71T^{2} \) |
| 73 | \( 1 - 1.96T + 73T^{2} \) |
| 79 | \( 1 + 8.26T + 79T^{2} \) |
| 83 | \( 1 - 8.22T + 83T^{2} \) |
| 89 | \( 1 + 1.59T + 89T^{2} \) |
| 97 | \( 1 + 7.65T + 97T^{2} \) |
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
−8.064193530351072321859868637200, −7.64469029261495378506390575335, −6.62573136728794995047987836069, −5.91230140882933932128195274309, −5.48152511722215515906724727058, −4.35666730284202559321950876324, −3.88153720229287519560409589300, −2.39456772760247263183441787958, −1.54644615063061868542741810867, −0.936334764565943426754488473594,
0.936334764565943426754488473594, 1.54644615063061868542741810867, 2.39456772760247263183441787958, 3.88153720229287519560409589300, 4.35666730284202559321950876324, 5.48152511722215515906724727058, 5.91230140882933932128195274309, 6.62573136728794995047987836069, 7.64469029261495378506390575335, 8.064193530351072321859868637200