L(s) = 1 | − 2-s + 2.08·3-s + 4-s + 0.616·5-s − 2.08·6-s + 4.73·7-s − 8-s + 1.34·9-s − 0.616·10-s + 0.598·11-s + 2.08·12-s + 6.36·13-s − 4.73·14-s + 1.28·15-s + 16-s + 2.56·17-s − 1.34·18-s − 2.23·19-s + 0.616·20-s + 9.86·21-s − 0.598·22-s − 7.35·23-s − 2.08·24-s − 4.62·25-s − 6.36·26-s − 3.45·27-s + 4.73·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.20·3-s + 0.5·4-s + 0.275·5-s − 0.850·6-s + 1.78·7-s − 0.353·8-s + 0.447·9-s − 0.194·10-s + 0.180·11-s + 0.601·12-s + 1.76·13-s − 1.26·14-s + 0.331·15-s + 0.250·16-s + 0.623·17-s − 0.316·18-s − 0.511·19-s + 0.137·20-s + 2.15·21-s − 0.127·22-s − 1.53·23-s − 0.425·24-s − 0.924·25-s − 1.24·26-s − 0.665·27-s + 0.894·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.102571691\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.102571691\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
good | 3 | \( 1 - 2.08T + 3T^{2} \) |
| 5 | \( 1 - 0.616T + 5T^{2} \) |
| 7 | \( 1 - 4.73T + 7T^{2} \) |
| 11 | \( 1 - 0.598T + 11T^{2} \) |
| 13 | \( 1 - 6.36T + 13T^{2} \) |
| 17 | \( 1 - 2.56T + 17T^{2} \) |
| 19 | \( 1 + 2.23T + 19T^{2} \) |
| 23 | \( 1 + 7.35T + 23T^{2} \) |
| 29 | \( 1 - 1.26T + 29T^{2} \) |
| 31 | \( 1 - 7.86T + 31T^{2} \) |
| 37 | \( 1 - 8.00T + 37T^{2} \) |
| 41 | \( 1 + 7.42T + 41T^{2} \) |
| 43 | \( 1 - 9.14T + 43T^{2} \) |
| 47 | \( 1 - 4.05T + 47T^{2} \) |
| 53 | \( 1 + 6.96T + 53T^{2} \) |
| 59 | \( 1 - 9.10T + 59T^{2} \) |
| 61 | \( 1 + 4.45T + 61T^{2} \) |
| 67 | \( 1 + 9.12T + 67T^{2} \) |
| 71 | \( 1 + 1.47T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 + 3.36T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 - 1.05T + 97T^{2} \) |
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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
−8.349718990077135166379548712520, −8.073068816898173206407558332141, −7.44838138087062607863761152485, −6.17212534430963599187087997891, −5.73162778950515665106883958729, −4.39705987303806592496807045290, −3.81396092892888389584737103573, −2.67371139136931621686252570318, −1.85523494910547528549920527404, −1.20124643674632341763698777136,
1.20124643674632341763698777136, 1.85523494910547528549920527404, 2.67371139136931621686252570318, 3.81396092892888389584737103573, 4.39705987303806592496807045290, 5.73162778950515665106883958729, 6.17212534430963599187087997891, 7.44838138087062607863761152485, 8.073068816898173206407558332141, 8.349718990077135166379548712520