| L(s) = 1 | + 0.585·5-s − 4.82·7-s − 2.82·11-s − 4.82·13-s + 3.41·17-s + 5.65·19-s + 5.41·23-s − 4.65·25-s + 1.75·29-s − 6.82·31-s − 2.82·35-s − 37-s + 11.6·41-s − 2.82·43-s − 5.65·47-s + 16.3·49-s + 12.8·53-s − 1.65·55-s + 4.24·59-s + 13.3·61-s − 2.82·65-s + 4.82·67-s − 8·71-s + 8·73-s + 13.6·77-s + 13.6·79-s + 0.485·83-s + ⋯ |
| L(s) = 1 | + 0.261·5-s − 1.82·7-s − 0.852·11-s − 1.33·13-s + 0.828·17-s + 1.29·19-s + 1.12·23-s − 0.931·25-s + 0.326·29-s − 1.22·31-s − 0.478·35-s − 0.164·37-s + 1.82·41-s − 0.431·43-s − 0.825·47-s + 2.33·49-s + 1.76·53-s − 0.223·55-s + 0.552·59-s + 1.70·61-s − 0.350·65-s + 0.589·67-s − 0.949·71-s + 0.936·73-s + 1.55·77-s + 1.53·79-s + 0.0532·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.138752644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.138752644\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 - 0.585T + 5T^{2} \) |
| 7 | \( 1 + 4.82T + 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 17 | \( 1 - 3.41T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 - 5.41T + 23T^{2} \) |
| 29 | \( 1 - 1.75T + 29T^{2} \) |
| 31 | \( 1 + 6.82T + 31T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 + 2.82T + 43T^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 - 4.24T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 - 4.82T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 0.485T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−9.173600693372147938682937301986, −7.922810962306191120465967733074, −7.25758607612397101262315506497, −6.69793747069327430172809301451, −5.53413353430735119121269351325, −5.30817654674783078546262725132, −3.86713175420322982562519143319, −3.05810897881563679597266210612, −2.39776509711746329883742241275, −0.63611373220484911698743267493,
0.63611373220484911698743267493, 2.39776509711746329883742241275, 3.05810897881563679597266210612, 3.86713175420322982562519143319, 5.30817654674783078546262725132, 5.53413353430735119121269351325, 6.69793747069327430172809301451, 7.25758607612397101262315506497, 7.922810962306191120465967733074, 9.173600693372147938682937301986
