| L(s) = 1 | − 2-s − 1.89·3-s + 4-s − 5-s + 1.89·6-s + 0.730·7-s − 8-s + 0.600·9-s + 10-s + 2.37·11-s − 1.89·12-s + 13-s − 0.730·14-s + 1.89·15-s + 16-s − 6.53·17-s − 0.600·18-s + 7.54·19-s − 20-s − 1.38·21-s − 2.37·22-s + 6.56·23-s + 1.89·24-s + 25-s − 26-s + 4.55·27-s + 0.730·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.09·3-s + 0.5·4-s − 0.447·5-s + 0.774·6-s + 0.276·7-s − 0.353·8-s + 0.200·9-s + 0.316·10-s + 0.714·11-s − 0.547·12-s + 0.277·13-s − 0.195·14-s + 0.489·15-s + 0.250·16-s − 1.58·17-s − 0.141·18-s + 1.73·19-s − 0.223·20-s − 0.302·21-s − 0.505·22-s + 1.36·23-s + 0.387·24-s + 0.200·25-s − 0.196·26-s + 0.876·27-s + 0.138·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7302943788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7302943788\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 3 | \( 1 + 1.89T + 3T^{2} \) |
| 7 | \( 1 - 0.730T + 7T^{2} \) |
| 11 | \( 1 - 2.37T + 11T^{2} \) |
| 17 | \( 1 + 6.53T + 17T^{2} \) |
| 19 | \( 1 - 7.54T + 19T^{2} \) |
| 23 | \( 1 - 6.56T + 23T^{2} \) |
| 29 | \( 1 + 3.52T + 29T^{2} \) |
| 37 | \( 1 + 8.37T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 - 3.79T + 47T^{2} \) |
| 53 | \( 1 - 9.62T + 53T^{2} \) |
| 59 | \( 1 + 0.122T + 59T^{2} \) |
| 61 | \( 1 + 9.26T + 61T^{2} \) |
| 67 | \( 1 - 7.43T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 + 1.92T + 73T^{2} \) |
| 79 | \( 1 + 0.500T + 79T^{2} \) |
| 83 | \( 1 + 15.2T + 83T^{2} \) |
| 89 | \( 1 - 5.13T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−8.650565760554473660706464595889, −7.58730359441740607753234434628, −6.94697607915599351795803168392, −6.43910997204712213827718034254, −5.46334290483316742159076731584, −4.91085139473446448695256975175, −3.86530950280238146765819589524, −2.89362506315866736495185415231, −1.56637703897281132304148759242, −0.60301920714356262184543684183,
0.60301920714356262184543684183, 1.56637703897281132304148759242, 2.89362506315866736495185415231, 3.86530950280238146765819589524, 4.91085139473446448695256975175, 5.46334290483316742159076731584, 6.43910997204712213827718034254, 6.94697607915599351795803168392, 7.58730359441740607753234434628, 8.650565760554473660706464595889
