| L(s) = 1 | − 2-s + 3.42·3-s + 4-s + 5-s − 3.42·6-s − 0.184·7-s − 8-s + 8.75·9-s − 10-s + 11-s + 3.42·12-s − 1.86·13-s + 0.184·14-s + 3.42·15-s + 16-s + 1.38·17-s − 8.75·18-s + 2.95·19-s + 20-s − 0.632·21-s − 22-s + 4.77·23-s − 3.42·24-s + 25-s + 1.86·26-s + 19.7·27-s − 0.184·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.97·3-s + 0.5·4-s + 0.447·5-s − 1.39·6-s − 0.0697·7-s − 0.353·8-s + 2.91·9-s − 0.316·10-s + 0.301·11-s + 0.989·12-s − 0.517·13-s + 0.0493·14-s + 0.885·15-s + 0.250·16-s + 0.335·17-s − 2.06·18-s + 0.677·19-s + 0.223·20-s − 0.138·21-s − 0.213·22-s + 0.996·23-s − 0.699·24-s + 0.200·25-s + 0.366·26-s + 3.79·27-s − 0.0348·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.674542925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.674542925\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 - 3.42T + 3T^{2} \) |
| 7 | \( 1 + 0.184T + 7T^{2} \) |
| 13 | \( 1 + 1.86T + 13T^{2} \) |
| 17 | \( 1 - 1.38T + 17T^{2} \) |
| 19 | \( 1 - 2.95T + 19T^{2} \) |
| 23 | \( 1 - 4.77T + 23T^{2} \) |
| 29 | \( 1 + 2.99T + 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 + 8.37T + 37T^{2} \) |
| 41 | \( 1 + 4.20T + 41T^{2} \) |
| 47 | \( 1 + 8.11T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 - 0.668T + 59T^{2} \) |
| 61 | \( 1 + 1.18T + 61T^{2} \) |
| 67 | \( 1 + 7.72T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 + 17.0T + 79T^{2} \) |
| 83 | \( 1 - 9.61T + 83T^{2} \) |
| 89 | \( 1 - 9.80T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−8.328990593885953530039722723615, −7.83343428763882529186371877556, −7.03566917274230610897449591236, −6.62487934862621037005992031948, −5.26733877295042773249929999349, −4.38381640002133078442721804037, −3.29489864256822046371568388319, −2.89608203071631052191299811747, −1.93731653990956197826984733401, −1.18589860329243067506047997612,
1.18589860329243067506047997612, 1.93731653990956197826984733401, 2.89608203071631052191299811747, 3.29489864256822046371568388319, 4.38381640002133078442721804037, 5.26733877295042773249929999349, 6.62487934862621037005992031948, 7.03566917274230610897449591236, 7.83343428763882529186371877556, 8.328990593885953530039722723615
