| L(s) = 1 | + 1.74·2-s − 2.89·3-s + 1.06·4-s + 0.790·5-s − 5.06·6-s − 1.64·8-s + 5.36·9-s + 1.38·10-s + 2.51·11-s − 3.06·12-s − 4.59·13-s − 2.28·15-s − 4.99·16-s − 17-s + 9.38·18-s + 0.869·19-s + 0.838·20-s + 4.40·22-s − 6.01·23-s + 4.75·24-s − 4.37·25-s − 8.03·26-s − 6.84·27-s − 7.62·29-s − 3.99·30-s − 6.34·31-s − 5.45·32-s + ⋯ |
| L(s) = 1 | + 1.23·2-s − 1.67·3-s + 0.530·4-s + 0.353·5-s − 2.06·6-s − 0.580·8-s + 1.78·9-s + 0.437·10-s + 0.758·11-s − 0.885·12-s − 1.27·13-s − 0.590·15-s − 1.24·16-s − 0.242·17-s + 2.21·18-s + 0.199·19-s + 0.187·20-s + 0.938·22-s − 1.25·23-s + 0.969·24-s − 0.875·25-s − 1.57·26-s − 1.31·27-s − 1.41·29-s − 0.730·30-s − 1.14·31-s − 0.964·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - 1.74T + 2T^{2} \) |
| 3 | \( 1 + 2.89T + 3T^{2} \) |
| 5 | \( 1 - 0.790T + 5T^{2} \) |
| 11 | \( 1 - 2.51T + 11T^{2} \) |
| 13 | \( 1 + 4.59T + 13T^{2} \) |
| 19 | \( 1 - 0.869T + 19T^{2} \) |
| 23 | \( 1 + 6.01T + 23T^{2} \) |
| 29 | \( 1 + 7.62T + 29T^{2} \) |
| 31 | \( 1 + 6.34T + 31T^{2} \) |
| 37 | \( 1 - 9.87T + 37T^{2} \) |
| 41 | \( 1 - 4.58T + 41T^{2} \) |
| 43 | \( 1 + 12.1T + 43T^{2} \) |
| 47 | \( 1 + 4.67T + 47T^{2} \) |
| 53 | \( 1 + 0.380T + 53T^{2} \) |
| 59 | \( 1 + 8.52T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 + 2.57T + 67T^{2} \) |
| 71 | \( 1 + 3.34T + 71T^{2} \) |
| 73 | \( 1 + 2.68T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 - 4.37T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−9.861285206907712775815507824576, −9.367427062173686598077050966596, −7.67467603413936791830099417129, −6.66865750113347967568866215713, −6.00026956881991100071968579538, −5.37112270500394460494064037658, −4.58651417742389739598756044840, −3.74095880844742451973781744525, −2.02253854192454319255145192920, 0,
2.02253854192454319255145192920, 3.74095880844742451973781744525, 4.58651417742389739598756044840, 5.37112270500394460494064037658, 6.00026956881991100071968579538, 6.66865750113347967568866215713, 7.67467603413936791830099417129, 9.367427062173686598077050966596, 9.861285206907712775815507824576
