L(s) = 1 | + 2-s + 2.50·3-s + 4-s − 5-s + 2.50·6-s − 4.61·7-s + 8-s + 3.27·9-s − 10-s − 1.79·11-s + 2.50·12-s + 1.88·13-s − 4.61·14-s − 2.50·15-s + 16-s − 4.96·17-s + 3.27·18-s − 3.39·19-s − 20-s − 11.5·21-s − 1.79·22-s − 3.88·23-s + 2.50·24-s + 25-s + 1.88·26-s + 0.699·27-s − 4.61·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.44·3-s + 0.5·4-s − 0.447·5-s + 1.02·6-s − 1.74·7-s + 0.353·8-s + 1.09·9-s − 0.316·10-s − 0.541·11-s + 0.723·12-s + 0.523·13-s − 1.23·14-s − 0.646·15-s + 0.250·16-s − 1.20·17-s + 0.772·18-s − 0.778·19-s − 0.223·20-s − 2.52·21-s − 0.382·22-s − 0.810·23-s + 0.511·24-s + 0.200·25-s + 0.370·26-s + 0.134·27-s − 0.871·28-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.50T + 3T^{2} \) |
| 7 | \( 1 + 4.61T + 7T^{2} \) |
| 11 | \( 1 + 1.79T + 11T^{2} \) |
| 13 | \( 1 - 1.88T + 13T^{2} \) |
| 17 | \( 1 + 4.96T + 17T^{2} \) |
| 19 | \( 1 + 3.39T + 19T^{2} \) |
| 23 | \( 1 + 3.88T + 23T^{2} \) |
| 29 | \( 1 - 1.77T + 29T^{2} \) |
| 31 | \( 1 - 7.02T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 0.471T + 41T^{2} \) |
| 43 | \( 1 + 1.80T + 43T^{2} \) |
| 47 | \( 1 + 13.1T + 47T^{2} \) |
| 53 | \( 1 + 1.34T + 53T^{2} \) |
| 59 | \( 1 + 0.0911T + 59T^{2} \) |
| 61 | \( 1 + 2.62T + 61T^{2} \) |
| 67 | \( 1 + 6.15T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 7.44T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 - 4.85T + 89T^{2} \) |
| 97 | \( 1 + 4.19T + 97T^{2} \) |
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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
−8.266903931312855828701738472847, −7.27472828952639590269315532476, −6.57989895743711998963205384890, −6.10377858636807300471655403466, −4.79767592306548218172841151028, −3.96771630702237813834414024796, −3.36260875868962240362355885049, −2.79406383920401352230444910294, −1.95868328364000158724264878832, 0,
1.95868328364000158724264878832, 2.79406383920401352230444910294, 3.36260875868962240362355885049, 3.96771630702237813834414024796, 4.79767592306548218172841151028, 6.10377858636807300471655403466, 6.57989895743711998963205384890, 7.27472828952639590269315532476, 8.266903931312855828701738472847