L(s) = 1 | + 0.942·3-s + 5-s + 1.85·7-s − 2.11·9-s − 5.46·11-s − 1.73·13-s + 0.942·15-s + 7.05·17-s − 0.773·19-s + 1.75·21-s − 3.81·23-s + 25-s − 4.81·27-s − 5.07·29-s − 3.44·31-s − 5.14·33-s + 1.85·35-s + 10.1·37-s − 1.62·39-s + 10.9·41-s − 11.4·43-s − 2.11·45-s + 4.80·47-s − 3.54·49-s + 6.64·51-s − 11.9·53-s − 5.46·55-s + ⋯ |
L(s) = 1 | + 0.543·3-s + 0.447·5-s + 0.702·7-s − 0.704·9-s − 1.64·11-s − 0.479·13-s + 0.243·15-s + 1.71·17-s − 0.177·19-s + 0.382·21-s − 0.794·23-s + 0.200·25-s − 0.926·27-s − 0.943·29-s − 0.618·31-s − 0.896·33-s + 0.314·35-s + 1.66·37-s − 0.260·39-s + 1.71·41-s − 1.73·43-s − 0.314·45-s + 0.701·47-s − 0.505·49-s + 0.930·51-s − 1.63·53-s − 0.736·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 - T \) |
good | 3 | \( 1 - 0.942T + 3T^{2} \) |
| 7 | \( 1 - 1.85T + 7T^{2} \) |
| 11 | \( 1 + 5.46T + 11T^{2} \) |
| 13 | \( 1 + 1.73T + 13T^{2} \) |
| 17 | \( 1 - 7.05T + 17T^{2} \) |
| 19 | \( 1 + 0.773T + 19T^{2} \) |
| 23 | \( 1 + 3.81T + 23T^{2} \) |
| 29 | \( 1 + 5.07T + 29T^{2} \) |
| 31 | \( 1 + 3.44T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 - 4.80T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 - 9.86T + 59T^{2} \) |
| 61 | \( 1 + 2.97T + 61T^{2} \) |
| 67 | \( 1 - 1.49T + 67T^{2} \) |
| 71 | \( 1 + 4.03T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 + 6.78T + 79T^{2} \) |
| 83 | \( 1 + 8.89T + 83T^{2} \) |
| 89 | \( 1 - 9.29T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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
−7.939201015893486506339635928409, −7.35895995573256889954960801222, −6.04729385139709091640029046476, −5.54381012337474825777921385084, −5.02377005218618240146915506411, −3.98747139043815404272438732530, −2.96262838258283539336315037369, −2.48137323739428011561686108998, −1.51694611069969245260855497706, 0,
1.51694611069969245260855497706, 2.48137323739428011561686108998, 2.96262838258283539336315037369, 3.98747139043815404272438732530, 5.02377005218618240146915506411, 5.54381012337474825777921385084, 6.04729385139709091640029046476, 7.35895995573256889954960801222, 7.939201015893486506339635928409