| L(s) = 1 | + 4·11-s + 2·13-s + 2·17-s − 4·19-s + 8·23-s + 2·29-s − 8·31-s − 37-s − 10·41-s + 12·43-s − 7·49-s + 6·53-s + 4·59-s − 10·61-s − 4·67-s + 8·71-s + 6·73-s + 8·79-s + 4·83-s − 10·89-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s + 1.66·23-s + 0.371·29-s − 1.43·31-s − 0.164·37-s − 1.56·41-s + 1.82·43-s − 49-s + 0.824·53-s + 0.520·59-s − 1.28·61-s − 0.488·67-s + 0.949·71-s + 0.702·73-s + 0.900·79-s + 0.439·83-s − 1.05·89-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.52535613836821, −13.38945224654418, −12.58128873614222, −12.37549322529983, −11.82012007334898, −11.18231105039629, −10.86244148720128, −10.50039677554292, −9.715839699710807, −9.161551940200149, −9.032676736108392, −8.353719852357292, −7.899498690371934, −7.159815371033053, −6.733581507780580, −6.419219330918269, −5.651315108355289, −5.255131181077501, −4.566728387336953, −3.961480993704010, −3.560005118209722, −2.939364839303437, −2.191512464033444, −1.433530277856061, −1.020997274473000, 0,
1.020997274473000, 1.433530277856061, 2.191512464033444, 2.939364839303437, 3.560005118209722, 3.961480993704010, 4.566728387336953, 5.255131181077501, 5.651315108355289, 6.419219330918269, 6.733581507780580, 7.159815371033053, 7.899498690371934, 8.353719852357292, 9.032676736108392, 9.161551940200149, 9.715839699710807, 10.50039677554292, 10.86244148720128, 11.18231105039629, 11.82012007334898, 12.37549322529983, 12.58128873614222, 13.38945224654418, 13.52535613836821
