| L(s) = 1 | + 2-s − 2.73·3-s + 4-s − 3.73·5-s − 2.73·6-s − 3.98·7-s + 8-s + 4.50·9-s − 3.73·10-s + 2.12·11-s − 2.73·12-s + 0.839·13-s − 3.98·14-s + 10.2·15-s + 16-s + 2.17·17-s + 4.50·18-s − 5.54·19-s − 3.73·20-s + 10.9·21-s + 2.12·22-s + 3.87·23-s − 2.73·24-s + 8.94·25-s + 0.839·26-s − 4.12·27-s − 3.98·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.58·3-s + 0.5·4-s − 1.66·5-s − 1.11·6-s − 1.50·7-s + 0.353·8-s + 1.50·9-s − 1.18·10-s + 0.640·11-s − 0.790·12-s + 0.232·13-s − 1.06·14-s + 2.64·15-s + 0.250·16-s + 0.527·17-s + 1.06·18-s − 1.27·19-s − 0.834·20-s + 2.38·21-s + 0.453·22-s + 0.807·23-s − 0.559·24-s + 1.78·25-s + 0.164·26-s − 0.793·27-s − 0.752·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 5 | \( 1 + 3.73T + 5T^{2} \) |
| 7 | \( 1 + 3.98T + 7T^{2} \) |
| 11 | \( 1 - 2.12T + 11T^{2} \) |
| 13 | \( 1 - 0.839T + 13T^{2} \) |
| 17 | \( 1 - 2.17T + 17T^{2} \) |
| 19 | \( 1 + 5.54T + 19T^{2} \) |
| 23 | \( 1 - 3.87T + 23T^{2} \) |
| 29 | \( 1 + 0.330T + 29T^{2} \) |
| 31 | \( 1 - 0.561T + 31T^{2} \) |
| 37 | \( 1 - 3.52T + 37T^{2} \) |
| 41 | \( 1 + 3.39T + 41T^{2} \) |
| 43 | \( 1 - 9.88T + 43T^{2} \) |
| 47 | \( 1 - 6.13T + 47T^{2} \) |
| 53 | \( 1 + 13.9T + 53T^{2} \) |
| 59 | \( 1 - 5.52T + 59T^{2} \) |
| 61 | \( 1 - 0.0453T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 - 1.91T + 71T^{2} \) |
| 73 | \( 1 + 9.50T + 73T^{2} \) |
| 79 | \( 1 - 4.55T + 79T^{2} \) |
| 83 | \( 1 + 2.73T + 83T^{2} \) |
| 89 | \( 1 - 7.28T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.76201478484341104075721728792, −6.97515399911548454693724906844, −6.52515800349886377381507049519, −5.96683390083486909579459932843, −5.04116689884129357208797644138, −4.18105156540536543157763648533, −3.77998488104249835340865200035, −2.84471295126311127143548464746, −0.968572805334066319594765374816, 0,
0.968572805334066319594765374816, 2.84471295126311127143548464746, 3.77998488104249835340865200035, 4.18105156540536543157763648533, 5.04116689884129357208797644138, 5.96683390083486909579459932843, 6.52515800349886377381507049519, 6.97515399911548454693724906844, 7.76201478484341104075721728792
