| L(s) = 1 | + 4·2-s + 8·4-s + 17·5-s − 20·7-s + 68·10-s − 32·11-s − 80·14-s − 64·16-s + 13·17-s − 30·19-s + 136·20-s − 128·22-s − 78·23-s + 164·25-s − 160·28-s − 197·29-s + 74·31-s − 256·32-s + 52·34-s − 340·35-s + 227·37-s − 120·38-s − 165·41-s − 156·43-s − 256·44-s − 312·46-s − 162·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.52·5-s − 1.07·7-s + 2.15·10-s − 0.877·11-s − 1.52·14-s − 16-s + 0.185·17-s − 0.362·19-s + 1.52·20-s − 1.24·22-s − 0.707·23-s + 1.31·25-s − 1.07·28-s − 1.26·29-s + 0.428·31-s − 1.41·32-s + 0.262·34-s − 1.64·35-s + 1.00·37-s − 0.512·38-s − 0.628·41-s − 0.553·43-s − 0.877·44-s − 1.00·46-s − 0.502·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - p^{2} T + p^{3} T^{2} \) |
| 5 | \( 1 - 17 T + p^{3} T^{2} \) |
| 7 | \( 1 + 20 T + p^{3} T^{2} \) |
| 11 | \( 1 + 32 T + p^{3} T^{2} \) |
| 17 | \( 1 - 13 T + p^{3} T^{2} \) |
| 19 | \( 1 + 30 T + p^{3} T^{2} \) |
| 23 | \( 1 + 78 T + p^{3} T^{2} \) |
| 29 | \( 1 + 197 T + p^{3} T^{2} \) |
| 31 | \( 1 - 74 T + p^{3} T^{2} \) |
| 37 | \( 1 - 227 T + p^{3} T^{2} \) |
| 41 | \( 1 + 165 T + p^{3} T^{2} \) |
| 43 | \( 1 + 156 T + p^{3} T^{2} \) |
| 47 | \( 1 + 162 T + p^{3} T^{2} \) |
| 53 | \( 1 + 93 T + p^{3} T^{2} \) |
| 59 | \( 1 + 864 T + p^{3} T^{2} \) |
| 61 | \( 1 - 145 T + p^{3} T^{2} \) |
| 67 | \( 1 + 862 T + p^{3} T^{2} \) |
| 71 | \( 1 - 654 T + p^{3} T^{2} \) |
| 73 | \( 1 + 215 T + p^{3} T^{2} \) |
| 79 | \( 1 + 76 T + p^{3} T^{2} \) |
| 83 | \( 1 - 628 T + p^{3} T^{2} \) |
| 89 | \( 1 + 266 T + p^{3} T^{2} \) |
| 97 | \( 1 + 238 T + p^{3} T^{2} \) |
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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
−8.920136294650283049030384821464, −7.66162194189476310668250123646, −6.49430710776968043618448412964, −6.11280905173866511994350357653, −5.44826736949723099989523146247, −4.64009483917479849855926390300, −3.47599737417239054431810970550, −2.72190868951251984021587235607, −1.87110376828255682208425364792, 0,
1.87110376828255682208425364792, 2.72190868951251984021587235607, 3.47599737417239054431810970550, 4.64009483917479849855926390300, 5.44826736949723099989523146247, 6.11280905173866511994350357653, 6.49430710776968043618448412964, 7.66162194189476310668250123646, 8.920136294650283049030384821464
