L(s) = 1 | − 3-s − 1.73·5-s − 2.73·7-s + 9-s − 2.73·11-s + 1.73·15-s − 0.267·17-s + 4.73·19-s + 2.73·21-s + 8.19·23-s − 2.00·25-s − 27-s + 7.92·29-s + 1.46·31-s + 2.73·33-s + 4.73·35-s − 1.53·37-s − 5·41-s + 12.1·43-s − 1.73·45-s + 3.26·47-s + 0.464·49-s + 0.267·51-s − 7.92·53-s + 4.73·55-s − 4.73·57-s − 6.26·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.774·5-s − 1.03·7-s + 0.333·9-s − 0.823·11-s + 0.447·15-s − 0.0649·17-s + 1.08·19-s + 0.596·21-s + 1.70·23-s − 0.400·25-s − 0.192·27-s + 1.47·29-s + 0.262·31-s + 0.475·33-s + 0.799·35-s − 0.252·37-s − 0.780·41-s + 1.85·43-s − 0.258·45-s + 0.476·47-s + 0.0663·49-s + 0.0375·51-s − 1.08·53-s + 0.638·55-s − 0.626·57-s − 0.802·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 1.73T + 5T^{2} \) |
| 7 | \( 1 + 2.73T + 7T^{2} \) |
| 11 | \( 1 + 2.73T + 11T^{2} \) |
| 17 | \( 1 + 0.267T + 17T^{2} \) |
| 19 | \( 1 - 4.73T + 19T^{2} \) |
| 23 | \( 1 - 8.19T + 23T^{2} \) |
| 29 | \( 1 - 7.92T + 29T^{2} \) |
| 31 | \( 1 - 1.46T + 31T^{2} \) |
| 37 | \( 1 + 1.53T + 37T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 - 3.26T + 47T^{2} \) |
| 53 | \( 1 + 7.92T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 6.26T + 61T^{2} \) |
| 67 | \( 1 + 8.73T + 67T^{2} \) |
| 71 | \( 1 + 2.19T + 71T^{2} \) |
| 73 | \( 1 + 9.19T + 73T^{2} \) |
| 79 | \( 1 + 8.39T + 79T^{2} \) |
| 83 | \( 1 - 1.66T + 83T^{2} \) |
| 89 | \( 1 - 9.46T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−7.87514626430338417248699686501, −7.33834917040622634052807584106, −6.64166924215167527339639508484, −5.86844658257899849018064000441, −5.06001984708718948722267546863, −4.35744064203742990390267496171, −3.28244703044903084842834945063, −2.77467968877965928951046799271, −1.09206121676631554675627062125, 0,
1.09206121676631554675627062125, 2.77467968877965928951046799271, 3.28244703044903084842834945063, 4.35744064203742990390267496171, 5.06001984708718948722267546863, 5.86844658257899849018064000441, 6.64166924215167527339639508484, 7.33834917040622634052807584106, 7.87514626430338417248699686501