| L(s) = 1 | − 3-s − 2·9-s − 3·11-s + 3·17-s − 4·19-s − 25-s + 5·27-s + 3·33-s + 2·43-s + 49-s − 3·51-s + 4·57-s + 15·59-s − 10·67-s + 22·73-s + 75-s + 81-s + 9·83-s − 21·89-s + 7·97-s + 6·99-s + 21·107-s − 9·113-s − 13·121-s + 127-s − 2·129-s + 131-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s − 0.904·11-s + 0.727·17-s − 0.917·19-s − 1/5·25-s + 0.962·27-s + 0.522·33-s + 0.304·43-s + 1/7·49-s − 0.420·51-s + 0.529·57-s + 1.95·59-s − 1.22·67-s + 2.57·73-s + 0.115·75-s + 1/9·81-s + 0.987·83-s − 2.22·89-s + 0.710·97-s + 0.603·99-s + 2.03·107-s − 0.846·113-s − 1.18·121-s + 0.0887·127-s − 0.176·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 77 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 115 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 112 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.322201786856294976874201869905, −7.970807953511489366404662002571, −7.44859643232418348845391665012, −6.94724151524849483513555047214, −6.41320555031426794669019188123, −5.97271103635454119761734409045, −5.55459260693920646365650753250, −5.07731940485703338745602987825, −4.69646132799259128502969116376, −3.88467078358539469744994900792, −3.44936044645736744650900810410, −2.59890563251556268256292467639, −2.25680439095222856289170851916, −1.07775125640691409561558892351, 0,
1.07775125640691409561558892351, 2.25680439095222856289170851916, 2.59890563251556268256292467639, 3.44936044645736744650900810410, 3.88467078358539469744994900792, 4.69646132799259128502969116376, 5.07731940485703338745602987825, 5.55459260693920646365650753250, 5.97271103635454119761734409045, 6.41320555031426794669019188123, 6.94724151524849483513555047214, 7.44859643232418348845391665012, 7.970807953511489366404662002571, 8.322201786856294976874201869905
