L(s) = 1 | + 3-s − 4-s + 9-s − 12-s − 2·13-s + 16-s − 2·19-s + 25-s + 27-s − 36-s − 2·39-s + 48-s + 49-s + 2·52-s − 2·57-s + 61-s − 64-s + 2·73-s + 75-s + 2·76-s + 81-s − 2·97-s − 100-s − 2·103-s − 108-s + 2·109-s − 2·117-s + ⋯ |
L(s) = 1 | + 3-s − 4-s + 9-s − 12-s − 2·13-s + 16-s − 2·19-s + 25-s + 27-s − 36-s − 2·39-s + 48-s + 49-s + 2·52-s − 2·57-s + 61-s − 64-s + 2·73-s + 75-s + 2·76-s + 81-s − 2·97-s − 100-s − 2·103-s − 108-s + 2·109-s − 2·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7276643588\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7276643588\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 61 | \( 1 - T \) |
good | 2 | \( 1 + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( ( 1 + T )^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 + T )^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( ( 1 - T )^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( ( 1 + T )^{2} \) |
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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
−12.83087761726923673886646212862, −12.39195193396268868922684039319, −10.54369659007926209394245278328, −9.717903375476500356342712797967, −8.859482284770096334717823687742, −8.003018958203834663696043586735, −6.86752148016159565343191931229, −5.01302609419072710833841359669, −4.08407150800302480167473241350, −2.48484983306151831058638012197,
2.48484983306151831058638012197, 4.08407150800302480167473241350, 5.01302609419072710833841359669, 6.86752148016159565343191931229, 8.003018958203834663696043586735, 8.859482284770096334717823687742, 9.717903375476500356342712797967, 10.54369659007926209394245278328, 12.39195193396268868922684039319, 12.83087761726923673886646212862