L(s) = 1 | + 2-s + 0.384·3-s + 4-s + 5-s + 0.384·6-s + 0.951·7-s + 8-s − 2.85·9-s + 10-s + 11-s + 0.384·12-s − 0.320·13-s + 0.951·14-s + 0.384·15-s + 16-s + 7.41·17-s − 2.85·18-s − 1.54·19-s + 20-s + 0.366·21-s + 22-s + 3.49·23-s + 0.384·24-s + 25-s − 0.320·26-s − 2.25·27-s + 0.951·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.222·3-s + 0.5·4-s + 0.447·5-s + 0.157·6-s + 0.359·7-s + 0.353·8-s − 0.950·9-s + 0.316·10-s + 0.301·11-s + 0.111·12-s − 0.0888·13-s + 0.254·14-s + 0.0993·15-s + 0.250·16-s + 1.79·17-s − 0.672·18-s − 0.354·19-s + 0.223·20-s + 0.0798·21-s + 0.213·22-s + 0.729·23-s + 0.0785·24-s + 0.200·25-s − 0.0628·26-s − 0.433·27-s + 0.179·28-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
\(L(1)\) |
\(\approx\) |
\(3.948394035\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.948394035\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.384T + 3T^{2} \) |
| 7 | \( 1 - 0.951T + 7T^{2} \) |
| 13 | \( 1 + 0.320T + 13T^{2} \) |
| 17 | \( 1 - 7.41T + 17T^{2} \) |
| 19 | \( 1 + 1.54T + 19T^{2} \) |
| 23 | \( 1 - 3.49T + 23T^{2} \) |
| 29 | \( 1 - 6.61T + 29T^{2} \) |
| 31 | \( 1 + 6.17T + 31T^{2} \) |
| 37 | \( 1 + 1.82T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 47 | \( 1 + 9.72T + 47T^{2} \) |
| 53 | \( 1 - 8.60T + 53T^{2} \) |
| 59 | \( 1 - 5.29T + 59T^{2} \) |
| 61 | \( 1 - 7.08T + 61T^{2} \) |
| 67 | \( 1 - 7.23T + 67T^{2} \) |
| 71 | \( 1 - 4.45T + 71T^{2} \) |
| 73 | \( 1 + 7.08T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 - 2.86T + 83T^{2} \) |
| 89 | \( 1 - 8.64T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−8.276271462670720935237581517830, −7.53229692659077734792561545226, −6.75106735449229234753238152736, −5.89801000426091410132740732456, −5.41880572208193204425307517595, −4.69749190060381745598994613749, −3.62472261512117629901030010763, −3.01012385417195106350432608870, −2.12244228778325131825156845245, −1.02319833157380008975803505010,
1.02319833157380008975803505010, 2.12244228778325131825156845245, 3.01012385417195106350432608870, 3.62472261512117629901030010763, 4.69749190060381745598994613749, 5.41880572208193204425307517595, 5.89801000426091410132740732456, 6.75106735449229234753238152736, 7.53229692659077734792561545226, 8.276271462670720935237581517830