L(s) = 1 | − 2-s − 0.732·3-s + 4-s + 3.46·5-s + 0.732·6-s + 2·7-s − 8-s − 2.46·9-s − 3.46·10-s − 4.73·11-s − 0.732·12-s + 4.19·13-s − 2·14-s − 2.53·15-s + 16-s − 3.46·17-s + 2.46·18-s − 4·19-s + 3.46·20-s − 1.46·21-s + 4.73·22-s + 0.732·24-s + 6.99·25-s − 4.19·26-s + 4·27-s + 2·28-s − 8.19·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.422·3-s + 0.5·4-s + 1.54·5-s + 0.298·6-s + 0.755·7-s − 0.353·8-s − 0.821·9-s − 1.09·10-s − 1.42·11-s − 0.211·12-s + 1.16·13-s − 0.534·14-s − 0.654·15-s + 0.250·16-s − 0.840·17-s + 0.580·18-s − 0.917·19-s + 0.774·20-s − 0.319·21-s + 1.00·22-s + 0.149·24-s + 1.39·25-s − 0.822·26-s + 0.769·27-s + 0.377·28-s − 1.52·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7023833513\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7023833513\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 31 | \( 1 - T \) |
good | 3 | \( 1 + 0.732T + 3T^{2} \) |
| 5 | \( 1 - 3.46T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 13 | \( 1 - 4.19T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 8.19T + 29T^{2} \) |
| 37 | \( 1 + 0.196T + 37T^{2} \) |
| 41 | \( 1 - 9.46T + 41T^{2} \) |
| 43 | \( 1 + 6.19T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 1.26T + 53T^{2} \) |
| 59 | \( 1 + 9.46T + 59T^{2} \) |
| 61 | \( 1 - 4.19T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 8.39T + 97T^{2} \) |
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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
−15.10961136302316188302955279187, −13.83497903730883132305308545779, −12.92350812338711265224807687877, −11.10715475413325761600637129741, −10.64069823348915275996916219267, −9.199013831007284259368287306539, −8.146968785550988610817278437423, −6.29381172208304978165866464944, −5.35243085333046339083045647979, −2.21549807665649421000277949125,
2.21549807665649421000277949125, 5.35243085333046339083045647979, 6.29381172208304978165866464944, 8.146968785550988610817278437423, 9.199013831007284259368287306539, 10.64069823348915275996916219267, 11.10715475413325761600637129741, 12.92350812338711265224807687877, 13.83497903730883132305308545779, 15.10961136302316188302955279187