| L(s) = 1 | + 1.44·2-s + 2.75·3-s + 0.0832·4-s + 3.98·6-s − 1.64·7-s − 2.76·8-s + 4.61·9-s − 2.45·11-s + 0.229·12-s − 0.225·13-s − 2.37·14-s − 4.15·16-s − 5.45·17-s + 6.65·18-s + 4.16·19-s − 4.54·21-s − 3.54·22-s − 1.41·23-s − 7.63·24-s − 0.325·26-s + 4.44·27-s − 0.137·28-s − 0.147·29-s + 1.53·31-s − 0.470·32-s − 6.78·33-s − 7.86·34-s + ⋯ |
| L(s) = 1 | + 1.02·2-s + 1.59·3-s + 0.0416·4-s + 1.62·6-s − 0.622·7-s − 0.978·8-s + 1.53·9-s − 0.741·11-s + 0.0662·12-s − 0.0625·13-s − 0.635·14-s − 1.03·16-s − 1.32·17-s + 1.56·18-s + 0.955·19-s − 0.991·21-s − 0.756·22-s − 0.295·23-s − 1.55·24-s − 0.0638·26-s + 0.855·27-s − 0.0259·28-s − 0.0274·29-s + 0.275·31-s − 0.0831·32-s − 1.18·33-s − 1.34·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 - 1.44T + 2T^{2} \) |
| 3 | \( 1 - 2.75T + 3T^{2} \) |
| 7 | \( 1 + 1.64T + 7T^{2} \) |
| 11 | \( 1 + 2.45T + 11T^{2} \) |
| 13 | \( 1 + 0.225T + 13T^{2} \) |
| 17 | \( 1 + 5.45T + 17T^{2} \) |
| 19 | \( 1 - 4.16T + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 + 0.147T + 29T^{2} \) |
| 31 | \( 1 - 1.53T + 31T^{2} \) |
| 37 | \( 1 + 11.1T + 37T^{2} \) |
| 41 | \( 1 + 2.94T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + 7.51T + 47T^{2} \) |
| 53 | \( 1 + 14.4T + 53T^{2} \) |
| 59 | \( 1 - 4.70T + 59T^{2} \) |
| 61 | \( 1 - 2.54T + 61T^{2} \) |
| 67 | \( 1 - 9.13T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 - 5.54T + 73T^{2} \) |
| 79 | \( 1 - 15.2T + 79T^{2} \) |
| 83 | \( 1 + 8.01T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−7.950563370258857487387204913909, −6.81333699440001151340163647696, −6.48598814732648857174141345322, −5.19127419694220005062336299770, −4.84814397958311424981672440636, −3.67777245689765760600097297265, −3.43192961322278798724639450287, −2.65126347268320949511384992335, −1.88491951538953235948282116796, 0,
1.88491951538953235948282116796, 2.65126347268320949511384992335, 3.43192961322278798724639450287, 3.67777245689765760600097297265, 4.84814397958311424981672440636, 5.19127419694220005062336299770, 6.48598814732648857174141345322, 6.81333699440001151340163647696, 7.950563370258857487387204913909
