L(s) = 1 | − 2-s + 3-s + 4-s + 3·5-s − 6-s − 4·7-s − 8-s + 9-s − 3·10-s + 11-s + 12-s − 6·13-s + 4·14-s + 3·15-s + 16-s + 3·17-s − 18-s − 4·19-s + 3·20-s − 4·21-s − 22-s + 3·23-s − 24-s + 4·25-s + 6·26-s + 27-s − 4·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.301·11-s + 0.288·12-s − 1.66·13-s + 1.06·14-s + 0.774·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.917·19-s + 0.670·20-s − 0.872·21-s − 0.213·22-s + 0.625·23-s − 0.204·24-s + 4/5·25-s + 1.17·26-s + 0.192·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.694170468\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.694170468\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−8.782728730633005720277299303662, −7.60143391446533003860198031505, −7.08941869031664012547260331721, −6.29848726016125299358810754964, −5.79212233230231121401822620565, −4.72186047187610189713010214066, −3.53178144006286429023928058476, −2.59209975809965669138504489299, −2.21105980713761296932018108303, −0.78232382132814916172352385432,
0.78232382132814916172352385432, 2.21105980713761296932018108303, 2.59209975809965669138504489299, 3.53178144006286429023928058476, 4.72186047187610189713010214066, 5.79212233230231121401822620565, 6.29848726016125299358810754964, 7.08941869031664012547260331721, 7.60143391446533003860198031505, 8.782728730633005720277299303662