L(s) = 1 | + 3·3-s + 5·5-s + 9·9-s + 15·15-s + 14·17-s − 22·19-s + 34·23-s + 25·25-s + 27·27-s − 2·31-s + 45·45-s − 14·47-s + 49·49-s + 42·51-s − 86·53-s − 66·57-s + 118·61-s + 102·69-s + 75·75-s − 98·79-s + 81·81-s − 154·83-s + 70·85-s − 6·93-s − 110·95-s − 106·107-s + 22·109-s + ⋯ |
L(s) = 1 | + 3-s + 5-s + 9-s + 15-s + 0.823·17-s − 1.15·19-s + 1.47·23-s + 25-s + 27-s − 0.0645·31-s + 45-s − 0.297·47-s + 49-s + 0.823·51-s − 1.62·53-s − 1.15·57-s + 1.93·61-s + 1.47·69-s + 75-s − 1.24·79-s + 81-s − 1.85·83-s + 0.823·85-s − 0.0645·93-s − 1.15·95-s − 0.990·107-s + 0.201·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(3.535966122\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.535966122\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
good | 7 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( 1 - 14 T + p^{2} T^{2} \) |
| 19 | \( 1 + 22 T + p^{2} T^{2} \) |
| 23 | \( 1 - 34 T + p^{2} T^{2} \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 + 2 T + p^{2} T^{2} \) |
| 37 | \( ( 1 - p T )( 1 + p T ) \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( 1 + 14 T + p^{2} T^{2} \) |
| 53 | \( 1 + 86 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 - 118 T + p^{2} T^{2} \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( ( 1 - p T )( 1 + p T ) \) |
| 79 | \( 1 + 98 T + p^{2} T^{2} \) |
| 83 | \( 1 + 154 T + p^{2} T^{2} \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( ( 1 - p T )( 1 + p T ) \) |
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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
−9.778789210996436835986088731617, −8.971187231924498478035477835414, −8.375956563876835536686237490746, −7.29943019574404212091748337679, −6.54267615495860962534497481220, −5.46944831120731656055056893987, −4.46737294318499848525589293559, −3.26369749967234816523191262033, −2.36054040198612594023373609551, −1.26728988225055009152667195052,
1.26728988225055009152667195052, 2.36054040198612594023373609551, 3.26369749967234816523191262033, 4.46737294318499848525589293559, 5.46944831120731656055056893987, 6.54267615495860962534497481220, 7.29943019574404212091748337679, 8.375956563876835536686237490746, 8.971187231924498478035477835414, 9.778789210996436835986088731617