| L(s) = 1 | + 2·3-s + 3·9-s − 4·16-s − 5·25-s + 4·27-s + 10·31-s + 10·37-s − 14·47-s − 8·48-s − 9·49-s + 4·53-s − 6·59-s − 2·67-s + 6·71-s − 10·75-s + 5·81-s + 2·89-s + 20·93-s + 26·97-s + 26·103-s + 20·111-s − 14·113-s + 127-s + 131-s + 137-s + 139-s − 28·141-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s − 16-s − 25-s + 0.769·27-s + 1.79·31-s + 1.64·37-s − 2.04·47-s − 1.15·48-s − 9/7·49-s + 0.549·53-s − 0.781·59-s − 0.244·67-s + 0.712·71-s − 1.15·75-s + 5/9·81-s + 0.211·89-s + 2.07·93-s + 2.63·97-s + 2.56·103-s + 1.89·111-s − 1.31·113-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.35·141-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.308633975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.308633975\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 107 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 115 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 47 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 134 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
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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
−7.68794062247014462758087552021, −7.19742608968454064251358537662, −6.61289776364040204484804624675, −6.37250549919330927863485425232, −6.07004707392929795306499018708, −5.31495295155884607050472383115, −4.77290224364744032813673970877, −4.49640278582798122937392998994, −4.12324282443417323411558947963, −3.39380130341969957297479313713, −3.15317096274986611064477544023, −2.52114174269983034095386834162, −2.09484306499358959131555990430, −1.53825627706854947467848336561, −0.62330363422154110492253201967,
0.62330363422154110492253201967, 1.53825627706854947467848336561, 2.09484306499358959131555990430, 2.52114174269983034095386834162, 3.15317096274986611064477544023, 3.39380130341969957297479313713, 4.12324282443417323411558947963, 4.49640278582798122937392998994, 4.77290224364744032813673970877, 5.31495295155884607050472383115, 6.07004707392929795306499018708, 6.37250549919330927863485425232, 6.61289776364040204484804624675, 7.19742608968454064251358537662, 7.68794062247014462758087552021
