| L(s) = 1 | + 4·2-s − 14·3-s + 16·4-s − 56·6-s + 64·8-s + 115·9-s − 46·11-s − 224·12-s + 256·16-s − 574·17-s + 460·18-s + 434·19-s − 184·22-s − 896·24-s + 625·25-s − 476·27-s + 1.02e3·32-s + 644·33-s − 2.29e3·34-s + 1.84e3·36-s + 1.73e3·38-s − 1.24e3·41-s − 3.50e3·43-s − 736·44-s − 3.58e3·48-s + 2.40e3·49-s + 2.50e3·50-s + ⋯ |
| L(s) = 1 | + 2-s − 1.55·3-s + 4-s − 1.55·6-s + 8-s + 1.41·9-s − 0.380·11-s − 1.55·12-s + 16-s − 1.98·17-s + 1.41·18-s + 1.20·19-s − 0.380·22-s − 1.55·24-s + 25-s − 0.652·27-s + 32-s + 0.591·33-s − 1.98·34-s + 1.41·36-s + 1.20·38-s − 0.741·41-s − 1.89·43-s − 0.380·44-s − 1.55·48-s + 49-s + 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.102297224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.102297224\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 + 14 T + p^{4} T^{2} \) |
| 5 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 7 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 11 | \( 1 + 46 T + p^{4} T^{2} \) |
| 13 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 17 | \( 1 + 574 T + p^{4} T^{2} \) |
| 19 | \( 1 - 434 T + p^{4} T^{2} \) |
| 23 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 29 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 31 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 37 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 41 | \( 1 + 1246 T + p^{4} T^{2} \) |
| 43 | \( 1 + 3502 T + p^{4} T^{2} \) |
| 47 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 53 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 59 | \( 1 + 238 T + p^{4} T^{2} \) |
| 61 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 67 | \( 1 + 5134 T + p^{4} T^{2} \) |
| 71 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 73 | \( 1 - 9506 T + p^{4} T^{2} \) |
| 79 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 83 | \( 1 - 11186 T + p^{4} T^{2} \) |
| 89 | \( 1 - 5474 T + p^{4} T^{2} \) |
| 97 | \( 1 + 9982 T + p^{4} T^{2} \) |
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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
−21.87154101837720099307701417636, −20.22346760649820502538164194536, −18.09956007309929382048504864517, −16.66259023739137137276789694294, −15.47925433222068016089592852164, −13.32774194787948632114753463965, −11.87049028069603400213599156636, −10.73811589325915176885578282965, −6.69768504379387003991406451607, −5.01594537282362360148542834223,
5.01594537282362360148542834223, 6.69768504379387003991406451607, 10.73811589325915176885578282965, 11.87049028069603400213599156636, 13.32774194787948632114753463965, 15.47925433222068016089592852164, 16.66259023739137137276789694294, 18.09956007309929382048504864517, 20.22346760649820502538164194536, 21.87154101837720099307701417636
