| L(s) = 1 | + 2.41·2-s + 0.108·3-s + 3.83·4-s + 0.262·6-s − 4.32·7-s + 4.43·8-s − 2.98·9-s + 0.569·11-s + 0.416·12-s + 1.69·13-s − 10.4·14-s + 3.04·16-s + 5.86·17-s − 7.21·18-s − 5.53·19-s − 0.469·21-s + 1.37·22-s + 5.19·23-s + 0.481·24-s + 4.10·26-s − 0.650·27-s − 16.5·28-s − 6.22·29-s − 7.88·31-s − 1.51·32-s + 0.0618·33-s + 14.1·34-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 0.0627·3-s + 1.91·4-s + 0.107·6-s − 1.63·7-s + 1.56·8-s − 0.996·9-s + 0.171·11-s + 0.120·12-s + 0.471·13-s − 2.79·14-s + 0.761·16-s + 1.42·17-s − 1.70·18-s − 1.26·19-s − 0.102·21-s + 0.293·22-s + 1.08·23-s + 0.0983·24-s + 0.804·26-s − 0.125·27-s − 3.13·28-s − 1.15·29-s − 1.41·31-s − 0.268·32-s + 0.0107·33-s + 2.43·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 197 | \( 1 + T \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 3 | \( 1 - 0.108T + 3T^{2} \) |
| 7 | \( 1 + 4.32T + 7T^{2} \) |
| 11 | \( 1 - 0.569T + 11T^{2} \) |
| 13 | \( 1 - 1.69T + 13T^{2} \) |
| 17 | \( 1 - 5.86T + 17T^{2} \) |
| 19 | \( 1 + 5.53T + 19T^{2} \) |
| 23 | \( 1 - 5.19T + 23T^{2} \) |
| 29 | \( 1 + 6.22T + 29T^{2} \) |
| 31 | \( 1 + 7.88T + 31T^{2} \) |
| 37 | \( 1 - 1.10T + 37T^{2} \) |
| 41 | \( 1 + 9.40T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 + 8.42T + 59T^{2} \) |
| 61 | \( 1 - 6.37T + 61T^{2} \) |
| 67 | \( 1 + 6.51T + 67T^{2} \) |
| 71 | \( 1 + 7.33T + 71T^{2} \) |
| 73 | \( 1 - 5.18T + 73T^{2} \) |
| 79 | \( 1 - 4.03T + 79T^{2} \) |
| 83 | \( 1 - 0.207T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 4.64T + 97T^{2} \) |
| 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
−7.59809275917408156431307921464, −6.67472762637841374795987684651, −6.38772283905704275912085618869, −5.56749242092252513458235425005, −5.14920311575760079752845772305, −3.88799800365167992888136505784, −3.38255645099353260644283097242, −2.98290558562584655278720098973, −1.84054327203077838054366611041, 0,
1.84054327203077838054366611041, 2.98290558562584655278720098973, 3.38255645099353260644283097242, 3.88799800365167992888136505784, 5.14920311575760079752845772305, 5.56749242092252513458235425005, 6.38772283905704275912085618869, 6.67472762637841374795987684651, 7.59809275917408156431307921464
