| L(s) = 1 | + 2-s + 4-s + 4·7-s + 8-s − 5·13-s + 4·14-s + 16-s − 3·17-s − 19-s + 3·23-s − 5·26-s + 4·28-s − 9·29-s − 4·31-s + 32-s − 3·34-s − 5·37-s − 38-s − 6·41-s + 10·43-s + 3·46-s + 6·47-s + 9·49-s − 5·52-s − 53-s + 4·56-s − 9·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.51·7-s + 0.353·8-s − 1.38·13-s + 1.06·14-s + 1/4·16-s − 0.727·17-s − 0.229·19-s + 0.625·23-s − 0.980·26-s + 0.755·28-s − 1.67·29-s − 0.718·31-s + 0.176·32-s − 0.514·34-s − 0.821·37-s − 0.162·38-s − 0.937·41-s + 1.52·43-s + 0.442·46-s + 0.875·47-s + 9/7·49-s − 0.693·52-s − 0.137·53-s + 0.534·56-s − 1.18·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 53 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−15.45791739001789, −15.05630849348143, −14.49232219716164, −14.33304952081056, −13.58334085891536, −13.00093337065977, −12.48050300170855, −11.92117764992456, −11.41307800739283, −10.84825860327789, −10.58210199013610, −9.574014427831437, −9.131738820590221, −8.363003394588184, −7.822009980868456, −7.105904278197186, −6.964622373837412, −5.745551386234735, −5.382271193054327, −4.786204602382569, −4.277252290380888, −3.596322984495189, −2.541357519739281, −2.094985303420911, −1.348907806961379, 0,
1.348907806961379, 2.094985303420911, 2.541357519739281, 3.596322984495189, 4.277252290380888, 4.786204602382569, 5.382271193054327, 5.745551386234735, 6.964622373837412, 7.105904278197186, 7.822009980868456, 8.363003394588184, 9.131738820590221, 9.574014427831437, 10.58210199013610, 10.84825860327789, 11.41307800739283, 11.92117764992456, 12.48050300170855, 13.00093337065977, 13.58334085891536, 14.33304952081056, 14.49232219716164, 15.05630849348143, 15.45791739001789
