L(s) = 1 | + 2·5-s − 2·9-s − 8·13-s − 4·17-s + 3·25-s + 4·29-s + 8·37-s + 12·41-s − 4·45-s + 2·49-s − 16·53-s + 4·61-s − 16·65-s + 12·73-s − 5·81-s − 8·85-s + 28·89-s − 24·97-s + 20·101-s − 4·109-s − 24·113-s + 16·117-s − 6·121-s + 4·125-s + 127-s + 131-s + 137-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 2/3·9-s − 2.21·13-s − 0.970·17-s + 3/5·25-s + 0.742·29-s + 1.31·37-s + 1.87·41-s − 0.596·45-s + 2/7·49-s − 2.19·53-s + 0.512·61-s − 1.98·65-s + 1.40·73-s − 5/9·81-s − 0.867·85-s + 2.96·89-s − 2.43·97-s + 1.99·101-s − 0.383·109-s − 2.25·113-s + 1.47·117-s − 0.545·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.67065823780629293367938794534, −7.56186383761303726301658586747, −6.75748996467427670181333689729, −6.65071821165806094544770300594, −5.89615972636884024734860870557, −5.81913226127580309415137377006, −4.89054562104167346793520173269, −4.88072322821672453742216928381, −4.37339781441300626459767562156, −3.58012982806994395167456994162, −2.82540491064399701707550154132, −2.40090939476093597229953098979, −2.21004067490187676234569474220, −1.08126566627806166434984305189, 0,
1.08126566627806166434984305189, 2.21004067490187676234569474220, 2.40090939476093597229953098979, 2.82540491064399701707550154132, 3.58012982806994395167456994162, 4.37339781441300626459767562156, 4.88072322821672453742216928381, 4.89054562104167346793520173269, 5.81913226127580309415137377006, 5.89615972636884024734860870557, 6.65071821165806094544770300594, 6.75748996467427670181333689729, 7.56186383761303726301658586747, 7.67065823780629293367938794534