| L(s) = 1 | − 4.46i·5-s − 2.64·7-s − 4.46i·11-s − 7.34i·17-s + 8.64·19-s − 2.88i·23-s − 14.9·25-s + 8.29·31-s + 11.8i·35-s − 3.93·37-s − 2.88i·41-s + 7.00·49-s − 19.9·55-s + 10.2i·71-s + 11.8i·77-s + ⋯ |
| L(s) = 1 | − 1.99i·5-s − 0.999·7-s − 1.34i·11-s − 1.78i·17-s + 1.98·19-s − 0.601i·23-s − 2.98·25-s + 1.48·31-s + 1.99i·35-s − 0.647·37-s − 0.450i·41-s + 49-s − 2.68·55-s + 1.21i·71-s + 1.34i·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.370620880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.370620880\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.64T \) |
| good | 5 | \( 1 + 4.46iT - 5T^{2} \) |
| 11 | \( 1 + 4.46iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 7.34iT - 17T^{2} \) |
| 19 | \( 1 - 8.64T + 19T^{2} \) |
| 23 | \( 1 + 2.88iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 8.29T + 31T^{2} \) |
| 37 | \( 1 + 3.93T + 37T^{2} \) |
| 41 | \( 1 + 2.88iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 10.2iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 14.9iT - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.487323626116947342117478671312, −7.76318741483986121551814589487, −6.84722759712054968191380909931, −5.79266815760565350125540030818, −5.31735849532340681210473135984, −4.58811966439921181975386114843, −3.53534354442055969036632678668, −2.72740356856827842876243764385, −1.04163315293928335231985275052, −0.50976318418740276799249696482,
1.72525877072554997608860338842, 2.79298328559921511687295869755, 3.38065324834146286888349856525, 4.14742963629888266037061920740, 5.50528189964945061613132990178, 6.29282709920406511579249613749, 6.82924484118791492034023121028, 7.42225497212835482530391002317, 8.084799033788969294687871250925, 9.477455818545599849117290735165
