L(s) = 1 | + 1.75·2-s + 3-s + 1.08·4-s + 1.72·5-s + 1.75·6-s − 1.67·7-s − 1.60·8-s + 9-s + 3.02·10-s + 2.50·11-s + 1.08·12-s + 13-s − 2.94·14-s + 1.72·15-s − 4.99·16-s + 3.11·17-s + 1.75·18-s + 1.37·19-s + 1.86·20-s − 1.67·21-s + 4.39·22-s + 5.86·23-s − 1.60·24-s − 2.03·25-s + 1.75·26-s + 27-s − 1.81·28-s + ⋯ |
L(s) = 1 | + 1.24·2-s + 0.577·3-s + 0.542·4-s + 0.770·5-s + 0.717·6-s − 0.633·7-s − 0.567·8-s + 0.333·9-s + 0.956·10-s + 0.754·11-s + 0.313·12-s + 0.277·13-s − 0.786·14-s + 0.444·15-s − 1.24·16-s + 0.756·17-s + 0.414·18-s + 0.315·19-s + 0.418·20-s − 0.365·21-s + 0.936·22-s + 1.22·23-s − 0.327·24-s − 0.406·25-s + 0.344·26-s + 0.192·27-s − 0.343·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.073196355\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.073196355\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 - 1.75T + 2T^{2} \) |
| 5 | \( 1 - 1.72T + 5T^{2} \) |
| 7 | \( 1 + 1.67T + 7T^{2} \) |
| 11 | \( 1 - 2.50T + 11T^{2} \) |
| 17 | \( 1 - 3.11T + 17T^{2} \) |
| 19 | \( 1 - 1.37T + 19T^{2} \) |
| 23 | \( 1 - 5.86T + 23T^{2} \) |
| 29 | \( 1 - 5.01T + 29T^{2} \) |
| 31 | \( 1 - 3.40T + 31T^{2} \) |
| 37 | \( 1 + 7.73T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 - 7.27T + 43T^{2} \) |
| 47 | \( 1 + 5.25T + 47T^{2} \) |
| 53 | \( 1 + 1.69T + 53T^{2} \) |
| 59 | \( 1 - 9.53T + 59T^{2} \) |
| 61 | \( 1 + 0.366T + 61T^{2} \) |
| 67 | \( 1 + 9.52T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 - 1.01T + 73T^{2} \) |
| 79 | \( 1 + 3.32T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 - 9.23T + 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−8.603654881546629973783427795680, −7.53745222382095279112302797003, −6.65091115050351784636479779335, −6.17247241553881480367613880084, −5.42080895488685562568509488982, −4.64291628816145792161560041715, −3.75188049124894147066597179243, −3.15712832858446285433624821094, −2.38956599226600742036083035526, −1.11375557669447362032382216839,
1.11375557669447362032382216839, 2.38956599226600742036083035526, 3.15712832858446285433624821094, 3.75188049124894147066597179243, 4.64291628816145792161560041715, 5.42080895488685562568509488982, 6.17247241553881480367613880084, 6.65091115050351784636479779335, 7.53745222382095279112302797003, 8.603654881546629973783427795680