| L(s) = 1 | − 0.302·3-s − 1.30·5-s − 4.60·7-s − 2.90·9-s + 1.30·11-s + 2.30·13-s + 0.394·15-s − 6·17-s + 2·19-s + 1.39·21-s + 6.90·23-s − 3.30·25-s + 1.78·27-s − 6.90·29-s − 3.30·31-s − 0.394·33-s + 6·35-s − 37-s − 0.697·39-s − 0.908·41-s − 6.60·43-s + 3.78·45-s + 2.60·47-s + 14.2·49-s + 1.81·51-s + 6·53-s − 1.69·55-s + ⋯ |
| L(s) = 1 | − 0.174·3-s − 0.582·5-s − 1.74·7-s − 0.969·9-s + 0.392·11-s + 0.638·13-s + 0.101·15-s − 1.45·17-s + 0.458·19-s + 0.304·21-s + 1.44·23-s − 0.660·25-s + 0.344·27-s − 1.28·29-s − 0.593·31-s − 0.0686·33-s + 1.01·35-s − 0.164·37-s − 0.111·39-s − 0.141·41-s − 1.00·43-s + 0.564·45-s + 0.380·47-s + 2.03·49-s + 0.254·51-s + 0.824·53-s − 0.228·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7045758571\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7045758571\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 + 0.302T + 3T^{2} \) |
| 5 | \( 1 + 1.30T + 5T^{2} \) |
| 7 | \( 1 + 4.60T + 7T^{2} \) |
| 11 | \( 1 - 1.30T + 11T^{2} \) |
| 13 | \( 1 - 2.30T + 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 6.90T + 23T^{2} \) |
| 29 | \( 1 + 6.90T + 29T^{2} \) |
| 31 | \( 1 + 3.30T + 31T^{2} \) |
| 41 | \( 1 + 0.908T + 41T^{2} \) |
| 43 | \( 1 + 6.60T + 43T^{2} \) |
| 47 | \( 1 - 2.60T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 3.39T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 - 14.5T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 8.69T + 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 - 17.2T + 83T^{2} \) |
| 89 | \( 1 - 5.21T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−9.069209802411176238199506216307, −8.380031259325216394423208543422, −7.21671162425764860760013864727, −6.67328312580759042489787592924, −5.97357340456275159642676498569, −5.13168222989071166574785713754, −3.79424579121823740375501171518, −3.42950928286374688138209739524, −2.33349610642909562115570568698, −0.51328978480074329868698016162,
0.51328978480074329868698016162, 2.33349610642909562115570568698, 3.42950928286374688138209739524, 3.79424579121823740375501171518, 5.13168222989071166574785713754, 5.97357340456275159642676498569, 6.67328312580759042489787592924, 7.21671162425764860760013864727, 8.380031259325216394423208543422, 9.069209802411176238199506216307
