| L(s) = 1 | + 3.56·5-s − 1.56·11-s − 0.438·13-s − 17-s − 4.68·19-s − 2.43·23-s + 7.68·25-s − 8.24·29-s − 3.12·31-s + 5.12·37-s + 3.56·41-s + 4.68·43-s − 11.1·47-s − 7·49-s + 12.2·53-s − 5.56·55-s − 7.12·59-s − 9.12·61-s − 1.56·65-s + 4·67-s − 6.24·71-s − 12.2·73-s + 9.36·79-s + 0.876·83-s − 3.56·85-s + 1.12·89-s − 16.6·95-s + ⋯ |
| L(s) = 1 | + 1.59·5-s − 0.470·11-s − 0.121·13-s − 0.242·17-s − 1.07·19-s − 0.508·23-s + 1.53·25-s − 1.53·29-s − 0.560·31-s + 0.842·37-s + 0.556·41-s + 0.714·43-s − 1.62·47-s − 49-s + 1.68·53-s − 0.749·55-s − 0.927·59-s − 1.16·61-s − 0.193·65-s + 0.488·67-s − 0.741·71-s − 1.43·73-s + 1.05·79-s + 0.0962·83-s − 0.386·85-s + 0.119·89-s − 1.71·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 3.56T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 13 | \( 1 + 0.438T + 13T^{2} \) |
| 19 | \( 1 + 4.68T + 19T^{2} \) |
| 23 | \( 1 + 2.43T + 23T^{2} \) |
| 29 | \( 1 + 8.24T + 29T^{2} \) |
| 31 | \( 1 + 3.12T + 31T^{2} \) |
| 37 | \( 1 - 5.12T + 37T^{2} \) |
| 41 | \( 1 - 3.56T + 41T^{2} \) |
| 43 | \( 1 - 4.68T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + 7.12T + 59T^{2} \) |
| 61 | \( 1 + 9.12T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 6.24T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 - 9.36T + 79T^{2} \) |
| 83 | \( 1 - 0.876T + 83T^{2} \) |
| 89 | \( 1 - 1.12T + 89T^{2} \) |
| 97 | \( 1 + 2.87T + 97T^{2} \) |
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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
−7.32840980739345383142059780203, −6.40915306892093620376161310444, −6.02797001173122175031913424464, −5.40011740132329526261619039897, −4.68458018744703465604536205040, −3.83941928754823959055717449194, −2.76859821503143241849474354074, −2.13603206834015227783898084385, −1.48816015749874533837855189620, 0,
1.48816015749874533837855189620, 2.13603206834015227783898084385, 2.76859821503143241849474354074, 3.83941928754823959055717449194, 4.68458018744703465604536205040, 5.40011740132329526261619039897, 6.02797001173122175031913424464, 6.40915306892093620376161310444, 7.32840980739345383142059780203
