L(s) = 1 | − 0.766·5-s − 7-s − 5.54·11-s + 6.31·13-s − 7.15·17-s + 6.15·19-s + 7.17·23-s − 4.41·25-s − 3.16·29-s + 0.163·31-s + 0.766·35-s + 5.31·37-s − 0.865·41-s + 8.31·43-s + 2.36·47-s + 49-s − 4·53-s + 4.24·55-s + 2.36·59-s − 13.5·61-s − 4.83·65-s − 7.05·67-s + 7.86·71-s − 1.98·73-s + 5.54·77-s − 5.24·79-s − 12.5·83-s + ⋯ |
L(s) = 1 | − 0.342·5-s − 0.377·7-s − 1.67·11-s + 1.75·13-s − 1.73·17-s + 1.41·19-s + 1.49·23-s − 0.882·25-s − 0.587·29-s + 0.0293·31-s + 0.129·35-s + 0.873·37-s − 0.135·41-s + 1.26·43-s + 0.345·47-s + 0.142·49-s − 0.549·53-s + 0.573·55-s + 0.308·59-s − 1.73·61-s − 0.599·65-s − 0.861·67-s + 0.932·71-s − 0.232·73-s + 0.632·77-s − 0.590·79-s − 1.37·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + 0.766T + 5T^{2} \) |
| 11 | \( 1 + 5.54T + 11T^{2} \) |
| 13 | \( 1 - 6.31T + 13T^{2} \) |
| 17 | \( 1 + 7.15T + 17T^{2} \) |
| 19 | \( 1 - 6.15T + 19T^{2} \) |
| 23 | \( 1 - 7.17T + 23T^{2} \) |
| 29 | \( 1 + 3.16T + 29T^{2} \) |
| 31 | \( 1 - 0.163T + 31T^{2} \) |
| 37 | \( 1 - 5.31T + 37T^{2} \) |
| 41 | \( 1 + 0.865T + 41T^{2} \) |
| 43 | \( 1 - 8.31T + 43T^{2} \) |
| 47 | \( 1 - 2.36T + 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 - 2.36T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 + 7.05T + 67T^{2} \) |
| 71 | \( 1 - 7.86T + 71T^{2} \) |
| 73 | \( 1 + 1.98T + 73T^{2} \) |
| 79 | \( 1 + 5.24T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 + 4.86T + 89T^{2} \) |
| 97 | \( 1 - 3.26T + 97T^{2} \) |
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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
−7.64671538562603101610701110938, −7.17490275292980193885209775648, −6.19720813844065877036945559196, −5.66118250082932931228209481277, −4.82217021622557919878835899680, −4.01928868620050316680084386365, −3.16353609345142643925700885623, −2.50280131550290241217258821165, −1.22556150059834491286759426717, 0,
1.22556150059834491286759426717, 2.50280131550290241217258821165, 3.16353609345142643925700885623, 4.01928868620050316680084386365, 4.82217021622557919878835899680, 5.66118250082932931228209481277, 6.19720813844065877036945559196, 7.17490275292980193885209775648, 7.64671538562603101610701110938