L(s) = 1 | + 3-s − 0.381·5-s − 7-s + 9-s − 2.23·11-s − 0.145·13-s − 0.381·15-s − 5.47·17-s + 8.23·19-s − 21-s − 23-s − 4.85·25-s + 27-s − 2.70·29-s + 5·31-s − 2.23·33-s + 0.381·35-s − 0.527·37-s − 0.145·39-s + 8.70·41-s + 8.32·43-s − 0.381·45-s − 5.23·47-s + 49-s − 5.47·51-s − 3.61·53-s + 0.854·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.170·5-s − 0.377·7-s + 0.333·9-s − 0.674·11-s − 0.0404·13-s − 0.0986·15-s − 1.32·17-s + 1.88·19-s − 0.218·21-s − 0.208·23-s − 0.970·25-s + 0.192·27-s − 0.502·29-s + 0.898·31-s − 0.389·33-s + 0.0645·35-s − 0.0867·37-s − 0.0233·39-s + 1.35·41-s + 1.26·43-s − 0.0569·45-s − 0.763·47-s + 0.142·49-s − 0.766·51-s − 0.496·53-s + 0.115·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + 0.381T + 5T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 + 0.145T + 13T^{2} \) |
| 17 | \( 1 + 5.47T + 17T^{2} \) |
| 19 | \( 1 - 8.23T + 19T^{2} \) |
| 29 | \( 1 + 2.70T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 + 0.527T + 37T^{2} \) |
| 41 | \( 1 - 8.70T + 41T^{2} \) |
| 43 | \( 1 - 8.32T + 43T^{2} \) |
| 47 | \( 1 + 5.23T + 47T^{2} \) |
| 53 | \( 1 + 3.61T + 53T^{2} \) |
| 59 | \( 1 - 4.85T + 59T^{2} \) |
| 61 | \( 1 + 9.09T + 61T^{2} \) |
| 67 | \( 1 - 6.85T + 67T^{2} \) |
| 71 | \( 1 + 9.38T + 71T^{2} \) |
| 73 | \( 1 + 3.47T + 73T^{2} \) |
| 79 | \( 1 + 5.94T + 79T^{2} \) |
| 83 | \( 1 + 7.94T + 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 + 17.1T + 97T^{2} \) |
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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
−7.52412292762567520064767913132, −7.04755057467137186395322827729, −6.08917510380939203564963365080, −5.47565733543702613294703081691, −4.53851231323841241013272825803, −3.92451885214682736216044604619, −2.97386399381430501815190676565, −2.46745426239782070216495055304, −1.32580720577385318735724055540, 0,
1.32580720577385318735724055540, 2.46745426239782070216495055304, 2.97386399381430501815190676565, 3.92451885214682736216044604619, 4.53851231323841241013272825803, 5.47565733543702613294703081691, 6.08917510380939203564963365080, 7.04755057467137186395322827729, 7.52412292762567520064767913132