L(s) = 1 | − 2-s + 0.604·3-s + 4-s − 5-s − 0.604·6-s − 3.36·7-s − 8-s − 2.63·9-s + 10-s + 11-s + 0.604·12-s − 6.90·13-s + 3.36·14-s − 0.604·15-s + 16-s − 2.94·17-s + 2.63·18-s − 7.65·19-s − 20-s − 2.03·21-s − 22-s + 2.62·23-s − 0.604·24-s + 25-s + 6.90·26-s − 3.40·27-s − 3.36·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.348·3-s + 0.5·4-s − 0.447·5-s − 0.246·6-s − 1.26·7-s − 0.353·8-s − 0.878·9-s + 0.316·10-s + 0.301·11-s + 0.174·12-s − 1.91·13-s + 0.898·14-s − 0.156·15-s + 0.250·16-s − 0.714·17-s + 0.620·18-s − 1.75·19-s − 0.223·20-s − 0.443·21-s − 0.213·22-s + 0.546·23-s − 0.123·24-s + 0.200·25-s + 1.35·26-s − 0.655·27-s − 0.634·28-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
\(L(1)\) |
\(\approx\) |
\(0.2134650142\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2134650142\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.604T + 3T^{2} \) |
| 7 | \( 1 + 3.36T + 7T^{2} \) |
| 13 | \( 1 + 6.90T + 13T^{2} \) |
| 17 | \( 1 + 2.94T + 17T^{2} \) |
| 19 | \( 1 + 7.65T + 19T^{2} \) |
| 23 | \( 1 - 2.62T + 23T^{2} \) |
| 29 | \( 1 + 3.72T + 29T^{2} \) |
| 31 | \( 1 + 3.70T + 31T^{2} \) |
| 37 | \( 1 - 6.43T + 37T^{2} \) |
| 41 | \( 1 + 2.67T + 41T^{2} \) |
| 47 | \( 1 + 3.73T + 47T^{2} \) |
| 53 | \( 1 - 2.96T + 53T^{2} \) |
| 59 | \( 1 - 5.09T + 59T^{2} \) |
| 61 | \( 1 + 3.33T + 61T^{2} \) |
| 67 | \( 1 + 3.79T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 + 3.87T + 73T^{2} \) |
| 79 | \( 1 + 1.43T + 79T^{2} \) |
| 83 | \( 1 - 0.379T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 + 4.31T + 97T^{2} \) |
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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
−8.363526786363503706677670008834, −7.65188415567222399253860987501, −6.86796779109307521645364489810, −6.44674724336338532041547926654, −5.48416786813969766363914862815, −4.47435229231979388824254208433, −3.56211565337976464299875254565, −2.69559664504956850963775612185, −2.14150819434269808680532473736, −0.25352170802574020808757908790,
0.25352170802574020808757908790, 2.14150819434269808680532473736, 2.69559664504956850963775612185, 3.56211565337976464299875254565, 4.47435229231979388824254208433, 5.48416786813969766363914862815, 6.44674724336338532041547926654, 6.86796779109307521645364489810, 7.65188415567222399253860987501, 8.363526786363503706677670008834