L(s) = 1 | + 3-s − 2·7-s + 9-s − 2·13-s − 4·16-s − 12·19-s − 2·21-s + 25-s + 27-s + 4·31-s + 14·37-s − 2·39-s − 16·43-s − 4·48-s − 11·49-s − 12·57-s + 10·61-s − 2·63-s − 8·67-s − 12·73-s + 75-s − 6·79-s + 81-s + 4·91-s + 4·93-s − 22·97-s − 8·103-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.554·13-s − 16-s − 2.75·19-s − 0.436·21-s + 1/5·25-s + 0.192·27-s + 0.718·31-s + 2.30·37-s − 0.320·39-s − 2.43·43-s − 0.577·48-s − 1.57·49-s − 1.58·57-s + 1.28·61-s − 0.251·63-s − 0.977·67-s − 1.40·73-s + 0.115·75-s − 0.675·79-s + 1/9·81-s + 0.419·91-s + 0.414·93-s − 2.23·97-s − 0.788·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114075 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114075 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
show more | | |
show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.348864711978766678634703610950, −8.599163171775301233119270744200, −8.333107995317450905784282789603, −7.997850874518894594264893149223, −6.93279974611194334514178729187, −6.92169485416648038433791572415, −6.28215256608024799923334344773, −5.86662536743360060967332495720, −4.75380031772719504345197730313, −4.52648172003467775210616179348, −3.93328920262448493706980455054, −3.02709857296590648239002053477, −2.53020079052262607160250914109, −1.78483850539275904424541360817, 0,
1.78483850539275904424541360817, 2.53020079052262607160250914109, 3.02709857296590648239002053477, 3.93328920262448493706980455054, 4.52648172003467775210616179348, 4.75380031772719504345197730313, 5.86662536743360060967332495720, 6.28215256608024799923334344773, 6.92169485416648038433791572415, 6.93279974611194334514178729187, 7.997850874518894594264893149223, 8.333107995317450905784282789603, 8.599163171775301233119270744200, 9.348864711978766678634703610950