| L(s) = 1 | + 2-s − 1.32·3-s + 4-s − 0.966·5-s − 1.32·6-s + 1.91·7-s + 8-s − 1.24·9-s − 0.966·10-s − 11-s − 1.32·12-s − 2.50·13-s + 1.91·14-s + 1.27·15-s + 16-s + 7.28·17-s − 1.24·18-s − 1.15·19-s − 0.966·20-s − 2.53·21-s − 22-s − 4.79·23-s − 1.32·24-s − 4.06·25-s − 2.50·26-s + 5.62·27-s + 1.91·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.764·3-s + 0.5·4-s − 0.432·5-s − 0.540·6-s + 0.725·7-s + 0.353·8-s − 0.415·9-s − 0.305·10-s − 0.301·11-s − 0.382·12-s − 0.695·13-s + 0.512·14-s + 0.330·15-s + 0.250·16-s + 1.76·17-s − 0.294·18-s − 0.266·19-s − 0.216·20-s − 0.554·21-s − 0.213·22-s − 0.999·23-s − 0.270·24-s − 0.813·25-s − 0.492·26-s + 1.08·27-s + 0.362·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 + 1.32T + 3T^{2} \) |
| 5 | \( 1 + 0.966T + 5T^{2} \) |
| 7 | \( 1 - 1.91T + 7T^{2} \) |
| 13 | \( 1 + 2.50T + 13T^{2} \) |
| 17 | \( 1 - 7.28T + 17T^{2} \) |
| 19 | \( 1 + 1.15T + 19T^{2} \) |
| 23 | \( 1 + 4.79T + 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 - 8.75T + 31T^{2} \) |
| 37 | \( 1 + 1.15T + 37T^{2} \) |
| 41 | \( 1 - 2.43T + 41T^{2} \) |
| 43 | \( 1 - 0.934T + 43T^{2} \) |
| 47 | \( 1 + 6.71T + 47T^{2} \) |
| 53 | \( 1 + 3.08T + 53T^{2} \) |
| 59 | \( 1 - 5.97T + 59T^{2} \) |
| 61 | \( 1 + 2.38T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 + 7.76T + 71T^{2} \) |
| 73 | \( 1 + 0.0133T + 73T^{2} \) |
| 79 | \( 1 + 4.62T + 79T^{2} \) |
| 83 | \( 1 - 2.54T + 83T^{2} \) |
| 89 | \( 1 + 3.60T + 89T^{2} \) |
| 97 | \( 1 + 1.71T + 97T^{2} \) |
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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
−7.86870036873019572625473044984, −7.34908543960588364063034058085, −6.24515393112424454635582981973, −5.71386600875260735138797710158, −5.08434222154590049072084803371, −4.40095432202297038960407749167, −3.48997631748513275651765916453, −2.57942720336529204496321509663, −1.43470504769911336074880210149, 0,
1.43470504769911336074880210149, 2.57942720336529204496321509663, 3.48997631748513275651765916453, 4.40095432202297038960407749167, 5.08434222154590049072084803371, 5.71386600875260735138797710158, 6.24515393112424454635582981973, 7.34908543960588364063034058085, 7.86870036873019572625473044984
