| L(s) = 1 | + 5·2-s + 17·4-s + 5·5-s + 45·8-s + 25·10-s + 50·11-s + 20·13-s + 89·16-s + 10·17-s + 44·19-s + 85·20-s + 250·22-s + 120·23-s + 25·25-s + 100·26-s − 50·29-s − 108·31-s + 85·32-s + 50·34-s − 40·37-s + 220·38-s + 225·40-s − 400·41-s + 280·43-s + 850·44-s + 600·46-s + 280·47-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 17/8·4-s + 0.447·5-s + 1.98·8-s + 0.790·10-s + 1.37·11-s + 0.426·13-s + 1.39·16-s + 0.142·17-s + 0.531·19-s + 0.950·20-s + 2.42·22-s + 1.08·23-s + 1/5·25-s + 0.754·26-s − 0.320·29-s − 0.625·31-s + 0.469·32-s + 0.252·34-s − 0.177·37-s + 0.939·38-s + 0.889·40-s − 1.52·41-s + 0.993·43-s + 2.91·44-s + 1.92·46-s + 0.868·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(9.384217670\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.384217670\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 5 T + p^{3} T^{2} \) |
| 11 | \( 1 - 50 T + p^{3} T^{2} \) |
| 13 | \( 1 - 20 T + p^{3} T^{2} \) |
| 17 | \( 1 - 10 T + p^{3} T^{2} \) |
| 19 | \( 1 - 44 T + p^{3} T^{2} \) |
| 23 | \( 1 - 120 T + p^{3} T^{2} \) |
| 29 | \( 1 + 50 T + p^{3} T^{2} \) |
| 31 | \( 1 + 108 T + p^{3} T^{2} \) |
| 37 | \( 1 + 40 T + p^{3} T^{2} \) |
| 41 | \( 1 + 400 T + p^{3} T^{2} \) |
| 43 | \( 1 - 280 T + p^{3} T^{2} \) |
| 47 | \( 1 - 280 T + p^{3} T^{2} \) |
| 53 | \( 1 + 610 T + p^{3} T^{2} \) |
| 59 | \( 1 + 50 T + p^{3} T^{2} \) |
| 61 | \( 1 - 518 T + p^{3} T^{2} \) |
| 67 | \( 1 + 180 T + p^{3} T^{2} \) |
| 71 | \( 1 - 700 T + p^{3} T^{2} \) |
| 73 | \( 1 - 410 T + p^{3} T^{2} \) |
| 79 | \( 1 + 516 T + p^{3} T^{2} \) |
| 83 | \( 1 + 660 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1500 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1630 T + p^{3} T^{2} \) |
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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
−8.793187229701449563671972416293, −7.50718468072486797205505567386, −6.77969912521265096193876038299, −6.20288492557514016509218453120, −5.42897760661581883872963289222, −4.72916888173915500139682555637, −3.77026334308400371737709529672, −3.24071168678933781223549787064, −2.08966296465346072497848235385, −1.15853261629104863974978570178,
1.15853261629104863974978570178, 2.08966296465346072497848235385, 3.24071168678933781223549787064, 3.77026334308400371737709529672, 4.72916888173915500139682555637, 5.42897760661581883872963289222, 6.20288492557514016509218453120, 6.77969912521265096193876038299, 7.50718468072486797205505567386, 8.793187229701449563671972416293
