| L(s) = 1 | + 2-s − 2·3-s − 4-s − 2·6-s − 3·8-s + 9-s + 2·12-s + 2·13-s − 16-s + 18-s − 19-s + 4·23-s + 6·24-s + 2·26-s + 4·27-s + 6·29-s − 2·31-s + 5·32-s − 36-s − 2·37-s − 38-s − 4·39-s + 6·41-s − 8·43-s + 4·46-s − 12·47-s + 2·48-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s − 1/2·4-s − 0.816·6-s − 1.06·8-s + 1/3·9-s + 0.577·12-s + 0.554·13-s − 1/4·16-s + 0.235·18-s − 0.229·19-s + 0.834·23-s + 1.22·24-s + 0.392·26-s + 0.769·27-s + 1.11·29-s − 0.359·31-s + 0.883·32-s − 1/6·36-s − 0.328·37-s − 0.162·38-s − 0.640·39-s + 0.937·41-s − 1.21·43-s + 0.589·46-s − 1.75·47-s + 0.288·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.55393730793817, −14.08867733574160, −13.50639077592688, −13.07204703372062, −12.48959519348749, −12.28800068187613, −11.51748151609220, −11.19997966852383, −10.77011456024334, −9.977814898241169, −9.642759533710547, −8.869684214285641, −8.429859559127449, −7.929210990679031, −6.951602955627346, −6.425802920476720, −6.195632295188609, −5.398661635555524, −5.003423188580743, −4.655742403750855, −3.861684867807030, −3.277694265956484, −2.681528694410488, −1.564738800653047, −0.7737649796443954, 0,
0.7737649796443954, 1.564738800653047, 2.681528694410488, 3.277694265956484, 3.861684867807030, 4.655742403750855, 5.003423188580743, 5.398661635555524, 6.195632295188609, 6.425802920476720, 6.951602955627346, 7.929210990679031, 8.429859559127449, 8.869684214285641, 9.642759533710547, 9.977814898241169, 10.77011456024334, 11.19997966852383, 11.51748151609220, 12.28800068187613, 12.48959519348749, 13.07204703372062, 13.50639077592688, 14.08867733574160, 14.55393730793817
