| L(s) = 1 | − 2·2-s + 4·4-s + 9·5-s + 7·7-s − 8·8-s − 18·10-s + 18·11-s + 13·13-s − 14·14-s + 16·16-s − 60·17-s − 43·19-s + 36·20-s − 36·22-s − 9·23-s − 44·25-s − 26·26-s + 28·28-s + 249·29-s − 79·31-s − 32·32-s + 120·34-s + 63·35-s − 412·37-s + 86·38-s − 72·40-s − 222·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.804·5-s + 0.377·7-s − 0.353·8-s − 0.569·10-s + 0.493·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.856·17-s − 0.519·19-s + 0.402·20-s − 0.348·22-s − 0.0815·23-s − 0.351·25-s − 0.196·26-s + 0.188·28-s + 1.59·29-s − 0.457·31-s − 0.176·32-s + 0.605·34-s + 0.304·35-s − 1.83·37-s + 0.367·38-s − 0.284·40-s − 0.845·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
| 13 | \( 1 - p T \) |
| good | 5 | \( 1 - 9 T + p^{3} T^{2} \) |
| 11 | \( 1 - 18 T + p^{3} T^{2} \) |
| 17 | \( 1 + 60 T + p^{3} T^{2} \) |
| 19 | \( 1 + 43 T + p^{3} T^{2} \) |
| 23 | \( 1 + 9 T + p^{3} T^{2} \) |
| 29 | \( 1 - 249 T + p^{3} T^{2} \) |
| 31 | \( 1 + 79 T + p^{3} T^{2} \) |
| 37 | \( 1 + 412 T + p^{3} T^{2} \) |
| 41 | \( 1 + 222 T + p^{3} T^{2} \) |
| 43 | \( 1 + 295 T + p^{3} T^{2} \) |
| 47 | \( 1 + 411 T + p^{3} T^{2} \) |
| 53 | \( 1 - 237 T + p^{3} T^{2} \) |
| 59 | \( 1 - 384 T + p^{3} T^{2} \) |
| 61 | \( 1 + 466 T + p^{3} T^{2} \) |
| 67 | \( 1 + 1042 T + p^{3} T^{2} \) |
| 71 | \( 1 - 288 T + p^{3} T^{2} \) |
| 73 | \( 1 + 691 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1001 T + p^{3} T^{2} \) |
| 83 | \( 1 + 39 T + p^{3} T^{2} \) |
| 89 | \( 1 - 339 T + p^{3} T^{2} \) |
| 97 | \( 1 - 713 T + p^{3} T^{2} \) |
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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
−8.697008695388896598516016265403, −8.069012336792659061747742028949, −6.86428762846836928770629558661, −6.45520218725832774252810656534, −5.45942598159436203494874749018, −4.50698625979662281459054887464, −3.30775496598806657471276755372, −2.09863302386225072051392185439, −1.43613961360454166384426843883, 0,
1.43613961360454166384426843883, 2.09863302386225072051392185439, 3.30775496598806657471276755372, 4.50698625979662281459054887464, 5.45942598159436203494874749018, 6.45520218725832774252810656534, 6.86428762846836928770629558661, 8.069012336792659061747742028949, 8.697008695388896598516016265403
