| L(s) = 1 | + 4·2-s − 10·3-s + 16·4-s + 22·5-s − 40·6-s + 49·7-s + 64·8-s + 19·9-s + 88·10-s − 160·12-s + 310·13-s + 196·14-s − 220·15-s + 256·16-s + 76·18-s − 650·19-s + 352·20-s − 490·21-s − 958·23-s − 640·24-s − 141·25-s + 1.24e3·26-s + 620·27-s + 784·28-s − 880·30-s + 1.02e3·32-s + 1.07e3·35-s + ⋯ |
| L(s) = 1 | + 2-s − 1.11·3-s + 4-s + 0.879·5-s − 1.11·6-s + 7-s + 8-s + 0.234·9-s + 0.879·10-s − 1.11·12-s + 1.83·13-s + 14-s − 0.977·15-s + 16-s + 0.234·18-s − 1.80·19-s + 0.879·20-s − 1.11·21-s − 1.81·23-s − 1.11·24-s − 0.225·25-s + 1.83·26-s + 0.850·27-s + 28-s − 0.977·30-s + 32-s + 0.879·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.497420144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.497420144\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 + 10 T + p^{4} T^{2} \) |
| 5 | \( 1 - 22 T + p^{4} T^{2} \) |
| 11 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 13 | \( 1 - 310 T + p^{4} T^{2} \) |
| 17 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 19 | \( 1 + 650 T + p^{4} T^{2} \) |
| 23 | \( 1 + 958 T + p^{4} T^{2} \) |
| 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 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 43 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 47 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 53 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 59 | \( 1 + 1130 T + p^{4} T^{2} \) |
| 61 | \( 1 + 7370 T + p^{4} T^{2} \) |
| 67 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 71 | \( 1 - 2018 T + p^{4} T^{2} \) |
| 73 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 79 | \( 1 - 4418 T + p^{4} T^{2} \) |
| 83 | \( 1 + 13130 T + p^{4} T^{2} \) |
| 89 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 97 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
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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
−14.27027376408173576877041649052, −13.44882670381341517012043293786, −12.20380492557216170445531027422, −11.14977169147526628448540228431, −10.50187894850507326552391848994, −8.304588267827272361328028219710, −6.29612067724508244030814750116, −5.75858155481941668785050054566, −4.28546742263887474345422848360, −1.76978317574464188161100406240,
1.76978317574464188161100406240, 4.28546742263887474345422848360, 5.75858155481941668785050054566, 6.29612067724508244030814750116, 8.304588267827272361328028219710, 10.50187894850507326552391848994, 11.14977169147526628448540228431, 12.20380492557216170445531027422, 13.44882670381341517012043293786, 14.27027376408173576877041649052
