| L(s) = 1 | + 2-s + 4-s + 4.42·7-s + 8-s − 2.62·11-s − 5.80·13-s + 4.42·14-s + 16-s − 3.80·17-s + 19-s − 2.62·22-s − 2.62·23-s − 5.80·26-s + 4.42·28-s − 3.37·29-s − 4.42·31-s + 32-s − 3.80·34-s − 5.80·37-s + 38-s − 5.67·41-s + 10.9·43-s − 2.62·44-s − 2.62·46-s + 2.62·47-s + 12.6·49-s − 5.80·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.67·7-s + 0.353·8-s − 0.790·11-s − 1.61·13-s + 1.18·14-s + 0.250·16-s − 0.923·17-s + 0.229·19-s − 0.559·22-s − 0.546·23-s − 1.13·26-s + 0.836·28-s − 0.627·29-s − 0.795·31-s + 0.176·32-s − 0.652·34-s − 0.954·37-s + 0.162·38-s − 0.885·41-s + 1.67·43-s − 0.395·44-s − 0.386·46-s + 0.382·47-s + 1.80·49-s − 0.805·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 4.42T + 7T^{2} \) |
| 11 | \( 1 + 2.62T + 11T^{2} \) |
| 13 | \( 1 + 5.80T + 13T^{2} \) |
| 17 | \( 1 + 3.80T + 17T^{2} \) |
| 23 | \( 1 + 2.62T + 23T^{2} \) |
| 29 | \( 1 + 3.37T + 29T^{2} \) |
| 31 | \( 1 + 4.42T + 31T^{2} \) |
| 37 | \( 1 + 5.80T + 37T^{2} \) |
| 41 | \( 1 + 5.67T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 - 2.62T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 1.05T + 59T^{2} \) |
| 61 | \( 1 - 4.75T + 61T^{2} \) |
| 67 | \( 1 + 15.6T + 67T^{2} \) |
| 71 | \( 1 + 15.6T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 4.42T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 + 7.37T + 97T^{2} \) |
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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
−7.47328931425999704860138491765, −6.89822281176198746450474823625, −5.68941102478384567301592458254, −5.33616296523912045398659182247, −4.59003287182386651509907057275, −4.22268787188847659401012501087, −2.98159096782937508208356549140, −2.21215486267884727371759061022, −1.63857972936392363372430223311, 0,
1.63857972936392363372430223311, 2.21215486267884727371759061022, 2.98159096782937508208356549140, 4.22268787188847659401012501087, 4.59003287182386651509907057275, 5.33616296523912045398659182247, 5.68941102478384567301592458254, 6.89822281176198746450474823625, 7.47328931425999704860138491765
