L(s) = 1 | − 2-s + 1.51·3-s + 4-s − 5-s − 1.51·6-s + 1.25·7-s − 8-s − 0.692·9-s + 10-s − 4.69·11-s + 1.51·12-s + 3.82·13-s − 1.25·14-s − 1.51·15-s + 16-s − 0.345·17-s + 0.692·18-s + 5.84·19-s − 20-s + 1.91·21-s + 4.69·22-s + 4.93·23-s − 1.51·24-s + 25-s − 3.82·26-s − 5.60·27-s + 1.25·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.876·3-s + 0.5·4-s − 0.447·5-s − 0.620·6-s + 0.475·7-s − 0.353·8-s − 0.230·9-s + 0.316·10-s − 1.41·11-s + 0.438·12-s + 1.05·13-s − 0.336·14-s − 0.392·15-s + 0.250·16-s − 0.0837·17-s + 0.163·18-s + 1.34·19-s − 0.223·20-s + 0.417·21-s + 1.00·22-s + 1.02·23-s − 0.310·24-s + 0.200·25-s − 0.749·26-s − 1.07·27-s + 0.237·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.671028182\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.671028182\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 - 1.51T + 3T^{2} \) |
| 7 | \( 1 - 1.25T + 7T^{2} \) |
| 11 | \( 1 + 4.69T + 11T^{2} \) |
| 13 | \( 1 - 3.82T + 13T^{2} \) |
| 17 | \( 1 + 0.345T + 17T^{2} \) |
| 19 | \( 1 - 5.84T + 19T^{2} \) |
| 23 | \( 1 - 4.93T + 23T^{2} \) |
| 29 | \( 1 - 7.80T + 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 - 0.209T + 37T^{2} \) |
| 41 | \( 1 + 0.370T + 41T^{2} \) |
| 43 | \( 1 + 8.41T + 43T^{2} \) |
| 47 | \( 1 - 4.61T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 0.967T + 59T^{2} \) |
| 61 | \( 1 - 6.90T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 - 6.14T + 71T^{2} \) |
| 73 | \( 1 - 6.53T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 - 7.49T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 - 17.7T + 97T^{2} \) |
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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
−8.343304169925324294691307148631, −7.969119473723103428698986248460, −7.35555454049288050902565212174, −6.42890110184195560533034079266, −5.42233609237396752920717021417, −4.75381227905781501776489603965, −3.36335688671794989568292381502, −3.04042209616799358358554112340, −1.98035293513221766604615293711, −0.789707506582053095966475083833,
0.789707506582053095966475083833, 1.98035293513221766604615293711, 3.04042209616799358358554112340, 3.36335688671794989568292381502, 4.75381227905781501776489603965, 5.42233609237396752920717021417, 6.42890110184195560533034079266, 7.35555454049288050902565212174, 7.969119473723103428698986248460, 8.343304169925324294691307148631