| L(s) = 1 | + 2·2-s + 4·4-s − 29·7-s + 8·8-s + 57·11-s − 20·13-s − 58·14-s + 16·16-s − 72·17-s − 106·19-s + 114·22-s + 174·23-s − 40·26-s − 116·28-s + 210·29-s + 47·31-s + 32·32-s − 144·34-s − 2·37-s − 212·38-s + 6·41-s − 218·43-s + 228·44-s + 348·46-s + 474·47-s + 498·49-s − 80·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.56·7-s + 0.353·8-s + 1.56·11-s − 0.426·13-s − 1.10·14-s + 1/4·16-s − 1.02·17-s − 1.27·19-s + 1.10·22-s + 1.57·23-s − 0.301·26-s − 0.782·28-s + 1.34·29-s + 0.272·31-s + 0.176·32-s − 0.726·34-s − 0.00888·37-s − 0.905·38-s + 0.0228·41-s − 0.773·43-s + 0.781·44-s + 1.11·46-s + 1.47·47-s + 1.45·49-s − 0.213·52-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
| \(L(2)\) |
\(\approx\) |
\(2.806315171\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.806315171\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 29 T + p^{3} T^{2} \) |
| 11 | \( 1 - 57 T + p^{3} T^{2} \) |
| 13 | \( 1 + 20 T + p^{3} T^{2} \) |
| 17 | \( 1 + 72 T + p^{3} T^{2} \) |
| 19 | \( 1 + 106 T + p^{3} T^{2} \) |
| 23 | \( 1 - 174 T + p^{3} T^{2} \) |
| 29 | \( 1 - 210 T + p^{3} T^{2} \) |
| 31 | \( 1 - 47 T + p^{3} T^{2} \) |
| 37 | \( 1 + 2 T + p^{3} T^{2} \) |
| 41 | \( 1 - 6 T + p^{3} T^{2} \) |
| 43 | \( 1 + 218 T + p^{3} T^{2} \) |
| 47 | \( 1 - 474 T + p^{3} T^{2} \) |
| 53 | \( 1 - 81 T + p^{3} T^{2} \) |
| 59 | \( 1 + 84 T + p^{3} T^{2} \) |
| 61 | \( 1 - 56 T + p^{3} T^{2} \) |
| 67 | \( 1 - 142 T + p^{3} T^{2} \) |
| 71 | \( 1 + 360 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1159 T + p^{3} T^{2} \) |
| 79 | \( 1 + 160 T + p^{3} T^{2} \) |
| 83 | \( 1 - 735 T + p^{3} T^{2} \) |
| 89 | \( 1 - 954 T + p^{3} T^{2} \) |
| 97 | \( 1 + 191 T + p^{3} T^{2} \) |
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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
−9.161780828336821344698855908746, −8.685455917235905049056417166402, −7.17764297953052472931142367819, −6.52752474554886321814263710617, −6.25961345769970523884757874813, −4.86266679417908708059207413635, −4.05836861202191383988624256638, −3.21621306325591057911058787819, −2.25966356658620063742333018140, −0.73991930168104280397810642856,
0.73991930168104280397810642856, 2.25966356658620063742333018140, 3.21621306325591057911058787819, 4.05836861202191383988624256638, 4.86266679417908708059207413635, 6.25961345769970523884757874813, 6.52752474554886321814263710617, 7.17764297953052472931142367819, 8.685455917235905049056417166402, 9.161780828336821344698855908746
