| L(s) = 1 | + 2·2-s + 4·4-s − 11·7-s + 8·8-s − 36·11-s − 17·13-s − 22·14-s + 16·16-s + 12·17-s − 91·19-s − 72·22-s − 60·23-s − 34·26-s − 44·28-s − 276·29-s + 191·31-s + 32·32-s + 24·34-s − 254·37-s − 182·38-s − 60·41-s + 49·43-s − 144·44-s − 120·46-s + 600·47-s − 222·49-s − 68·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.593·7-s + 0.353·8-s − 0.986·11-s − 0.362·13-s − 0.419·14-s + 1/4·16-s + 0.171·17-s − 1.09·19-s − 0.697·22-s − 0.543·23-s − 0.256·26-s − 0.296·28-s − 1.76·29-s + 1.10·31-s + 0.176·32-s + 0.121·34-s − 1.12·37-s − 0.776·38-s − 0.228·41-s + 0.173·43-s − 0.493·44-s − 0.384·46-s + 1.86·47-s − 0.647·49-s − 0.181·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 11 T + p^{3} T^{2} \) |
| 11 | \( 1 + 36 T + p^{3} T^{2} \) |
| 13 | \( 1 + 17 T + p^{3} T^{2} \) |
| 17 | \( 1 - 12 T + p^{3} T^{2} \) |
| 19 | \( 1 + 91 T + p^{3} T^{2} \) |
| 23 | \( 1 + 60 T + p^{3} T^{2} \) |
| 29 | \( 1 + 276 T + p^{3} T^{2} \) |
| 31 | \( 1 - 191 T + p^{3} T^{2} \) |
| 37 | \( 1 + 254 T + p^{3} T^{2} \) |
| 41 | \( 1 + 60 T + p^{3} T^{2} \) |
| 43 | \( 1 - 49 T + p^{3} T^{2} \) |
| 47 | \( 1 - 600 T + p^{3} T^{2} \) |
| 53 | \( 1 + 612 T + p^{3} T^{2} \) |
| 59 | \( 1 + 744 T + p^{3} T^{2} \) |
| 61 | \( 1 - 167 T + p^{3} T^{2} \) |
| 67 | \( 1 - 457 T + p^{3} T^{2} \) |
| 71 | \( 1 + 588 T + p^{3} T^{2} \) |
| 73 | \( 1 - 970 T + p^{3} T^{2} \) |
| 79 | \( 1 - 164 T + p^{3} T^{2} \) |
| 83 | \( 1 + 696 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1248 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1099 T + p^{3} T^{2} \) |
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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
−10.37857610315694995129188180053, −9.472774419393355184405066152520, −8.245961374188239939869668143106, −7.33755234705688223690345601865, −6.31351973579512617301050039773, −5.43050451201831225638547262810, −4.34593689344460197818351222680, −3.19562311320940126138254562811, −2.07118589850210283768112708916, 0,
2.07118589850210283768112708916, 3.19562311320940126138254562811, 4.34593689344460197818351222680, 5.43050451201831225638547262810, 6.31351973579512617301050039773, 7.33755234705688223690345601865, 8.245961374188239939869668143106, 9.472774419393355184405066152520, 10.37857610315694995129188180053
