| L(s) = 1 | + 2·2-s + 4·4-s − 14·7-s + 8·8-s + 22·11-s + 30·13-s − 28·14-s + 16·16-s − 7·17-s − 81·19-s + 44·22-s − 151·23-s + 60·26-s − 56·28-s − 270·29-s − 113·31-s + 32·32-s − 14·34-s + 88·37-s − 162·38-s + 406·41-s + 442·43-s + 88·44-s − 302·46-s − 56·47-s − 147·49-s + 120·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s + 0.603·11-s + 0.640·13-s − 0.534·14-s + 1/4·16-s − 0.0998·17-s − 0.978·19-s + 0.426·22-s − 1.36·23-s + 0.452·26-s − 0.377·28-s − 1.72·29-s − 0.654·31-s + 0.176·32-s − 0.0706·34-s + 0.391·37-s − 0.691·38-s + 1.54·41-s + 1.56·43-s + 0.301·44-s − 0.967·46-s − 0.173·47-s − 3/7·49-s + 0.320·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 13 | \( 1 - 30 T + p^{3} T^{2} \) |
| 17 | \( 1 + 7 T + p^{3} T^{2} \) |
| 19 | \( 1 + 81 T + p^{3} T^{2} \) |
| 23 | \( 1 + 151 T + p^{3} T^{2} \) |
| 29 | \( 1 + 270 T + p^{3} T^{2} \) |
| 31 | \( 1 + 113 T + p^{3} T^{2} \) |
| 37 | \( 1 - 88 T + p^{3} T^{2} \) |
| 41 | \( 1 - 406 T + p^{3} T^{2} \) |
| 43 | \( 1 - 442 T + p^{3} T^{2} \) |
| 47 | \( 1 + 56 T + p^{3} T^{2} \) |
| 53 | \( 1 - 141 T + p^{3} T^{2} \) |
| 59 | \( 1 + 274 T + p^{3} T^{2} \) |
| 61 | \( 1 - 41 T + p^{3} T^{2} \) |
| 67 | \( 1 + 328 T + p^{3} T^{2} \) |
| 71 | \( 1 + 390 T + p^{3} T^{2} \) |
| 73 | \( 1 + 626 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1215 T + p^{3} T^{2} \) |
| 83 | \( 1 + 505 T + p^{3} T^{2} \) |
| 89 | \( 1 - 514 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1816 T + p^{3} T^{2} \) |
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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
−8.951182500006241751650274083806, −7.87531881122257678508796014730, −7.04992208817450423335664111098, −6.07555917368838925620106605672, −5.77723248813873307073006813276, −4.24043221482409557320090843944, −3.85410205094453489819810566230, −2.67700771241530920930609255320, −1.57971637766425711728907316732, 0,
1.57971637766425711728907316732, 2.67700771241530920930609255320, 3.85410205094453489819810566230, 4.24043221482409557320090843944, 5.77723248813873307073006813276, 6.07555917368838925620106605672, 7.04992208817450423335664111098, 7.87531881122257678508796014730, 8.951182500006241751650274083806
