L(s) = 1 | − 2-s − 0.443·3-s + 4-s − 2.41·5-s + 0.443·6-s − 1.05·7-s − 8-s − 2.80·9-s + 2.41·10-s − 2.84·11-s − 0.443·12-s + 4.85·13-s + 1.05·14-s + 1.07·15-s + 16-s − 3.20·17-s + 2.80·18-s + 0.893·19-s − 2.41·20-s + 0.468·21-s + 2.84·22-s − 7.97·23-s + 0.443·24-s + 0.842·25-s − 4.85·26-s + 2.57·27-s − 1.05·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.256·3-s + 0.5·4-s − 1.08·5-s + 0.181·6-s − 0.399·7-s − 0.353·8-s − 0.934·9-s + 0.764·10-s − 0.856·11-s − 0.128·12-s + 1.34·13-s + 0.282·14-s + 0.276·15-s + 0.250·16-s − 0.777·17-s + 0.660·18-s + 0.205·19-s − 0.540·20-s + 0.102·21-s + 0.605·22-s − 1.66·23-s + 0.0905·24-s + 0.168·25-s − 0.951·26-s + 0.495·27-s − 0.199·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4782924337\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4782924337\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 751 | \( 1 - T \) |
good | 3 | \( 1 + 0.443T + 3T^{2} \) |
| 5 | \( 1 + 2.41T + 5T^{2} \) |
| 7 | \( 1 + 1.05T + 7T^{2} \) |
| 11 | \( 1 + 2.84T + 11T^{2} \) |
| 13 | \( 1 - 4.85T + 13T^{2} \) |
| 17 | \( 1 + 3.20T + 17T^{2} \) |
| 19 | \( 1 - 0.893T + 19T^{2} \) |
| 23 | \( 1 + 7.97T + 23T^{2} \) |
| 29 | \( 1 - 1.29T + 29T^{2} \) |
| 31 | \( 1 + 3.66T + 31T^{2} \) |
| 37 | \( 1 - 4.76T + 37T^{2} \) |
| 41 | \( 1 - 3.84T + 41T^{2} \) |
| 43 | \( 1 - 7.81T + 43T^{2} \) |
| 47 | \( 1 - 0.0757T + 47T^{2} \) |
| 53 | \( 1 - 2.48T + 53T^{2} \) |
| 59 | \( 1 - 7.51T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 + 4.25T + 67T^{2} \) |
| 71 | \( 1 - 5.44T + 71T^{2} \) |
| 73 | \( 1 + 5.95T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 + 0.568T + 89T^{2} \) |
| 97 | \( 1 - 4.92T + 97T^{2} \) |
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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
−9.395483328819473036999742777273, −8.442443507368143119967845957964, −8.133324594311285926575990768126, −7.24973697247399862103091203369, −6.21621003226909270551592990653, −5.64236961724338992234480402902, −4.24659670916651818451133003702, −3.40546653613745809237954365926, −2.30362955575468785333379622385, −0.52381950739996515532185707081,
0.52381950739996515532185707081, 2.30362955575468785333379622385, 3.40546653613745809237954365926, 4.24659670916651818451133003702, 5.64236961724338992234480402902, 6.21621003226909270551592990653, 7.24973697247399862103091203369, 8.133324594311285926575990768126, 8.442443507368143119967845957964, 9.395483328819473036999742777273