| L(s) = 1 | − 2-s + 4-s − 8-s + 4·11-s + 13-s + 16-s + 2·17-s − 4·19-s − 4·22-s + 4·23-s − 26-s + 6·29-s − 4·31-s − 32-s − 2·34-s + 2·37-s + 4·38-s − 2·41-s + 4·43-s + 4·44-s − 4·46-s − 8·47-s + 52-s + 6·53-s − 6·58-s − 12·59-s + 10·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1.20·11-s + 0.277·13-s + 1/4·16-s + 0.485·17-s − 0.917·19-s − 0.852·22-s + 0.834·23-s − 0.196·26-s + 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.342·34-s + 0.328·37-s + 0.648·38-s − 0.312·41-s + 0.609·43-s + 0.603·44-s − 0.589·46-s − 1.16·47-s + 0.138·52-s + 0.824·53-s − 0.787·58-s − 1.56·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.642853726\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.642853726\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.53737496777547, −12.31276076187154, −11.69877920196727, −11.30099943438680, −10.86439699646415, −10.46770810282740, −9.889346173646180, −9.442115979455381, −9.053992310153691, −8.638059125798019, −8.079509987119464, −7.803415246627928, −6.888493964570096, −6.784520540482552, −6.336767087421552, −5.676744335780226, −5.219900746241132, −4.435221150341906, −4.113873643838527, −3.310188340540564, −3.067767079972510, −2.107567488523719, −1.780349721525213, −0.9511293926622184, −0.5973247065915828,
0.5973247065915828, 0.9511293926622184, 1.780349721525213, 2.107567488523719, 3.067767079972510, 3.310188340540564, 4.113873643838527, 4.435221150341906, 5.219900746241132, 5.676744335780226, 6.336767087421552, 6.784520540482552, 6.888493964570096, 7.803415246627928, 8.079509987119464, 8.638059125798019, 9.053992310153691, 9.442115979455381, 9.889346173646180, 10.46770810282740, 10.86439699646415, 11.30099943438680, 11.69877920196727, 12.31276076187154, 12.53737496777547
