L(s) = 1 | + 1.19·2-s + 1.42·3-s − 0.566·4-s + 1.70·6-s − 1.72·7-s − 3.07·8-s − 0.973·9-s − 6.17·11-s − 0.806·12-s + 2.79·13-s − 2.06·14-s − 2.54·16-s + 1.90·17-s − 1.16·18-s + 1.36·19-s − 2.45·21-s − 7.39·22-s + 1.31·23-s − 4.37·24-s + 3.34·26-s − 5.65·27-s + 0.977·28-s + 6.68·29-s + 9.60·31-s + 3.09·32-s − 8.78·33-s + 2.27·34-s + ⋯ |
L(s) = 1 | + 0.846·2-s + 0.821·3-s − 0.283·4-s + 0.695·6-s − 0.652·7-s − 1.08·8-s − 0.324·9-s − 1.86·11-s − 0.232·12-s + 0.775·13-s − 0.552·14-s − 0.636·16-s + 0.461·17-s − 0.274·18-s + 0.313·19-s − 0.535·21-s − 1.57·22-s + 0.274·23-s − 0.892·24-s + 0.656·26-s − 1.08·27-s + 0.184·28-s + 1.24·29-s + 1.72·31-s + 0.547·32-s − 1.52·33-s + 0.390·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.386133454\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.386133454\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
good | 2 | \( 1 - 1.19T + 2T^{2} \) |
| 3 | \( 1 - 1.42T + 3T^{2} \) |
| 7 | \( 1 + 1.72T + 7T^{2} \) |
| 11 | \( 1 + 6.17T + 11T^{2} \) |
| 13 | \( 1 - 2.79T + 13T^{2} \) |
| 17 | \( 1 - 1.90T + 17T^{2} \) |
| 19 | \( 1 - 1.36T + 19T^{2} \) |
| 23 | \( 1 - 1.31T + 23T^{2} \) |
| 29 | \( 1 - 6.68T + 29T^{2} \) |
| 31 | \( 1 - 9.60T + 31T^{2} \) |
| 37 | \( 1 - 2.69T + 37T^{2} \) |
| 41 | \( 1 - 0.155T + 41T^{2} \) |
| 43 | \( 1 - 1.63T + 43T^{2} \) |
| 47 | \( 1 + 0.0487T + 47T^{2} \) |
| 53 | \( 1 + 6.75T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 + 8.73T + 61T^{2} \) |
| 67 | \( 1 + 2.97T + 67T^{2} \) |
| 71 | \( 1 + 7.62T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 0.436T + 83T^{2} \) |
| 89 | \( 1 - 5.40T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−8.292991721206926563296198471850, −7.50651393570031885887401471504, −6.38160088833619620650667005816, −5.88858238291598782279083250479, −5.09402920988377154901706903168, −4.48578461621139087367188410169, −3.36202955675425574754317880720, −3.06280178078988962669608025366, −2.39315061357707766382962478121, −0.65645007980510875639172164791,
0.65645007980510875639172164791, 2.39315061357707766382962478121, 3.06280178078988962669608025366, 3.36202955675425574754317880720, 4.48578461621139087367188410169, 5.09402920988377154901706903168, 5.88858238291598782279083250479, 6.38160088833619620650667005816, 7.50651393570031885887401471504, 8.292991721206926563296198471850