L(s) = 1 | + 2-s − 3-s + 4-s − 3.00·5-s − 6-s − 0.900·7-s + 8-s + 9-s − 3.00·10-s − 6.30·11-s − 12-s − 3.45·13-s − 0.900·14-s + 3.00·15-s + 16-s + 17-s + 18-s + 6.45·19-s − 3.00·20-s + 0.900·21-s − 6.30·22-s − 2.88·23-s − 24-s + 4.05·25-s − 3.45·26-s − 27-s − 0.900·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.34·5-s − 0.408·6-s − 0.340·7-s + 0.353·8-s + 0.333·9-s − 0.951·10-s − 1.90·11-s − 0.288·12-s − 0.957·13-s − 0.240·14-s + 0.776·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s + 1.48·19-s − 0.672·20-s + 0.196·21-s − 1.34·22-s − 0.601·23-s − 0.204·24-s + 0.810·25-s − 0.676·26-s − 0.192·27-s − 0.170·28-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
\(L(1)\) |
\(\approx\) |
\(0.7781387341\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7781387341\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.00T + 5T^{2} \) |
| 7 | \( 1 + 0.900T + 7T^{2} \) |
| 11 | \( 1 + 6.30T + 11T^{2} \) |
| 13 | \( 1 + 3.45T + 13T^{2} \) |
| 19 | \( 1 - 6.45T + 19T^{2} \) |
| 23 | \( 1 + 2.88T + 23T^{2} \) |
| 29 | \( 1 - 2.91T + 29T^{2} \) |
| 31 | \( 1 + 1.87T + 31T^{2} \) |
| 37 | \( 1 + 8.38T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 + 4.63T + 43T^{2} \) |
| 47 | \( 1 + 9.49T + 47T^{2} \) |
| 53 | \( 1 - 6.66T + 53T^{2} \) |
| 61 | \( 1 - 1.98T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 9.78T + 71T^{2} \) |
| 73 | \( 1 + 4.23T + 73T^{2} \) |
| 79 | \( 1 - 5.62T + 79T^{2} \) |
| 83 | \( 1 - 16.5T + 83T^{2} \) |
| 89 | \( 1 - 9.46T + 89T^{2} \) |
| 97 | \( 1 - 7.99T + 97T^{2} \) |
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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
−7.906134638400736669436273413213, −7.28581212664282056704217303279, −6.80569110294145390810545442185, −5.68469679930362138374720409151, −5.03205800248013020627532942240, −4.74716062166608934575161857844, −3.47538087822207871716415768551, −3.18747777326527519009384079600, −2.01898313804184787912121790401, −0.40318287333234964180493139396,
0.40318287333234964180493139396, 2.01898313804184787912121790401, 3.18747777326527519009384079600, 3.47538087822207871716415768551, 4.74716062166608934575161857844, 5.03205800248013020627532942240, 5.68469679930362138374720409151, 6.80569110294145390810545442185, 7.28581212664282056704217303279, 7.906134638400736669436273413213