| L(s) = 1 | − 2.74·2-s + 3.09·3-s + 5.52·4-s − 0.455·5-s − 8.49·6-s + 3.01·7-s − 9.66·8-s + 6.58·9-s + 1.24·10-s + 2.78·11-s + 17.1·12-s − 13-s − 8.27·14-s − 1.40·15-s + 15.4·16-s − 5.86·17-s − 18.0·18-s − 4.73·19-s − 2.51·20-s + 9.34·21-s − 7.64·22-s + 23-s − 29.9·24-s − 4.79·25-s + 2.74·26-s + 11.0·27-s + 16.6·28-s + ⋯ |
| L(s) = 1 | − 1.93·2-s + 1.78·3-s + 2.76·4-s − 0.203·5-s − 3.46·6-s + 1.14·7-s − 3.41·8-s + 2.19·9-s + 0.395·10-s + 0.840·11-s + 4.93·12-s − 0.277·13-s − 2.21·14-s − 0.364·15-s + 3.86·16-s − 1.42·17-s − 4.25·18-s − 1.08·19-s − 0.562·20-s + 2.03·21-s − 1.62·22-s + 0.208·23-s − 6.10·24-s − 0.958·25-s + 0.537·26-s + 2.13·27-s + 3.14·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.089179720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.089179720\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 2.74T + 2T^{2} \) |
| 3 | \( 1 - 3.09T + 3T^{2} \) |
| 5 | \( 1 + 0.455T + 5T^{2} \) |
| 7 | \( 1 - 3.01T + 7T^{2} \) |
| 11 | \( 1 - 2.78T + 11T^{2} \) |
| 17 | \( 1 + 5.86T + 17T^{2} \) |
| 19 | \( 1 + 4.73T + 19T^{2} \) |
| 29 | \( 1 - 1.61T + 29T^{2} \) |
| 31 | \( 1 - 2.41T + 31T^{2} \) |
| 37 | \( 1 + 4.81T + 37T^{2} \) |
| 41 | \( 1 + 0.841T + 41T^{2} \) |
| 43 | \( 1 - 7.21T + 43T^{2} \) |
| 47 | \( 1 - 8.27T + 47T^{2} \) |
| 53 | \( 1 + 6.43T + 53T^{2} \) |
| 59 | \( 1 - 1.17T + 59T^{2} \) |
| 61 | \( 1 - 2.37T + 61T^{2} \) |
| 67 | \( 1 + 4.73T + 67T^{2} \) |
| 71 | \( 1 - 9.59T + 71T^{2} \) |
| 73 | \( 1 + 7.86T + 73T^{2} \) |
| 79 | \( 1 + 4.63T + 79T^{2} \) |
| 83 | \( 1 - 6.82T + 83T^{2} \) |
| 89 | \( 1 - 2.91T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−11.32930720277578211012821253471, −10.49953903029486188249637535830, −9.432705354321778685554661440031, −8.798156938729899780625702141516, −8.263195294362221010970044263772, −7.48874501731793352091163584991, −6.59717650025749830635322137283, −4.12947613195813248426964892815, −2.50311851816182267963869693083, −1.67045033152070213497533947942,
1.67045033152070213497533947942, 2.50311851816182267963869693083, 4.12947613195813248426964892815, 6.59717650025749830635322137283, 7.48874501731793352091163584991, 8.263195294362221010970044263772, 8.798156938729899780625702141516, 9.432705354321778685554661440031, 10.49953903029486188249637535830, 11.32930720277578211012821253471
