| L(s) = 1 | + 1.17·2-s − 1.90·3-s + 0.381·4-s − 5-s − 2.23·6-s − 0.726·8-s + 2.61·9-s − 1.17·10-s − 11-s − 0.726·12-s + 1.90·15-s − 1.23·16-s + 3.07·18-s + 0.618·19-s − 0.381·20-s − 1.17·22-s + 1.38·24-s + 25-s − 3.07·27-s + 1.61·29-s + 2.23·30-s + 0.618·31-s − 0.726·32-s + 1.90·33-s + 0.999·36-s + 0.726·38-s + 0.726·40-s + ⋯ |
| L(s) = 1 | + 1.17·2-s − 1.90·3-s + 0.381·4-s − 5-s − 2.23·6-s − 0.726·8-s + 2.61·9-s − 1.17·10-s − 11-s − 0.726·12-s + 1.90·15-s − 1.23·16-s + 3.07·18-s + 0.618·19-s − 0.381·20-s − 1.17·22-s + 1.38·24-s + 25-s − 3.07·27-s + 1.61·29-s + 2.23·30-s + 0.618·31-s − 0.726·32-s + 1.90·33-s + 0.999·36-s + 0.726·38-s + 0.726·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2255 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2255 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7334625824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7334625824\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 1.17T + T^{2} \) |
| 3 | \( 1 + 1.90T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 0.618T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 1.61T + T^{2} \) |
| 31 | \( 1 - 0.618T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - 1.17T + T^{2} \) |
| 59 | \( 1 + 1.61T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.17T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.90T + T^{2} \) |
| 79 | \( 1 - 1.61T + T^{2} \) |
| 83 | \( 1 - 1.90T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.396611294288520949086475385749, −8.184470457437893051005680736676, −7.32497335947891301799693129388, −6.59286135502712112862861290382, −5.85195516436989267884785029590, −5.12447505852288121892370423824, −4.63282143675936225174589130258, −3.92542896376914893197783310575, −2.77535716645227103636018265752, −0.74878159745897107922679280179,
0.74878159745897107922679280179, 2.77535716645227103636018265752, 3.92542896376914893197783310575, 4.63282143675936225174589130258, 5.12447505852288121892370423824, 5.85195516436989267884785029590, 6.59286135502712112862861290382, 7.32497335947891301799693129388, 8.184470457437893051005680736676, 9.396611294288520949086475385749
