| L(s) = 1 | − 3·3-s + 5-s + 6·9-s + 4·11-s − 13-s − 3·15-s − 6·17-s − 4·19-s − 23-s − 9·27-s − 6·29-s − 4·31-s − 12·33-s − 8·37-s + 3·39-s − 43-s + 6·45-s − 12·47-s + 7·49-s + 18·51-s + 2·53-s + 4·55-s + 12·57-s − 14·59-s − 7·61-s − 65-s + 12·67-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.447·5-s + 2·9-s + 1.20·11-s − 0.277·13-s − 0.774·15-s − 1.45·17-s − 0.917·19-s − 0.208·23-s − 1.73·27-s − 1.11·29-s − 0.718·31-s − 2.08·33-s − 1.31·37-s + 0.480·39-s − 0.152·43-s + 0.894·45-s − 1.75·47-s + 49-s + 2.52·51-s + 0.274·53-s + 0.539·55-s + 1.58·57-s − 1.82·59-s − 0.896·61-s − 0.124·65-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 14 T + 137 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 6 T - 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
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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
−8.942022955481218278795352075730, −8.588194045472877910327294407294, −7.82821087144715275082442639281, −7.60592205350298030259844254161, −6.90524849744935209455041566449, −6.84936937240749406726896047850, −6.33412562481949020165192355799, −6.23081391411209340975132850916, −5.62542049029418183647828306103, −5.39076333562346205935289017811, −4.67580295508113324508419483380, −4.66229086108823526196537601198, −3.90700456454890209013256337453, −3.82746585250110375491011833809, −2.93067235174357128493963874067, −2.22868860938577465041115696180, −1.54433392967760795808867545754, −1.46181414938195531533051402594, 0, 0,
1.46181414938195531533051402594, 1.54433392967760795808867545754, 2.22868860938577465041115696180, 2.93067235174357128493963874067, 3.82746585250110375491011833809, 3.90700456454890209013256337453, 4.66229086108823526196537601198, 4.67580295508113324508419483380, 5.39076333562346205935289017811, 5.62542049029418183647828306103, 6.23081391411209340975132850916, 6.33412562481949020165192355799, 6.84936937240749406726896047850, 6.90524849744935209455041566449, 7.60592205350298030259844254161, 7.82821087144715275082442639281, 8.588194045472877910327294407294, 8.942022955481218278795352075730
