| L(s) = 1 | + 2·7-s + 13-s + 19-s − 25-s − 2·31-s + 37-s + 43-s + 3·49-s + 4·61-s − 2·67-s + 73-s − 2·79-s + 2·91-s − 2·97-s + 103-s + 109-s − 121-s + 127-s + 131-s + 2·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 2·7-s + 13-s + 19-s − 25-s − 2·31-s + 37-s + 43-s + 3·49-s + 4·61-s − 2·67-s + 73-s − 2·79-s + 2·91-s − 2·97-s + 103-s + 109-s − 121-s + 127-s + 131-s + 2·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.845450047\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.845450047\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$ | \( ( 1 - T )^{4} \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
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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
−9.153338232986323186681088460622, −9.108688689345120411643750672502, −8.429828042820330762870312692366, −8.376110809271684023258551777402, −7.79530403988971392926624644083, −7.64073232880201584420174544260, −7.05323076797002470098049645351, −6.99001581039198255487883852803, −6.06049256396177347611625316284, −5.81569501731695088579930511292, −5.32608295124416406390418438935, −5.30844406603584785401115148180, −4.37248025714459566018551953218, −4.36660985697126908925200191638, −3.67283423915175459448753760639, −3.43488079466704275041015761409, −2.34228919283796378074577378497, −2.30551555332520783623695991816, −1.38506309809481737495560227578, −1.14599634746030754773680385690,
1.14599634746030754773680385690, 1.38506309809481737495560227578, 2.30551555332520783623695991816, 2.34228919283796378074577378497, 3.43488079466704275041015761409, 3.67283423915175459448753760639, 4.36660985697126908925200191638, 4.37248025714459566018551953218, 5.30844406603584785401115148180, 5.32608295124416406390418438935, 5.81569501731695088579930511292, 6.06049256396177347611625316284, 6.99001581039198255487883852803, 7.05323076797002470098049645351, 7.64073232880201584420174544260, 7.79530403988971392926624644083, 8.376110809271684023258551777402, 8.429828042820330762870312692366, 9.108688689345120411643750672502, 9.153338232986323186681088460622
