| L(s) = 1 | + 2·3-s + 5-s + 2·7-s + 3·9-s − 7·13-s + 2·15-s − 2·19-s + 4·21-s + 2·23-s − 8·25-s + 4·27-s − 12·29-s + 2·35-s − 22·37-s − 14·39-s + 6·41-s + 43-s + 3·45-s + 10·47-s + 3·49-s − 15·53-s − 4·57-s − 21·59-s − 3·61-s + 6·63-s − 7·65-s − 67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 0.755·7-s + 9-s − 1.94·13-s + 0.516·15-s − 0.458·19-s + 0.872·21-s + 0.417·23-s − 8/5·25-s + 0.769·27-s − 2.22·29-s + 0.338·35-s − 3.61·37-s − 2.24·39-s + 0.937·41-s + 0.152·43-s + 0.447·45-s + 1.45·47-s + 3/7·49-s − 2.06·53-s − 0.529·57-s − 2.73·59-s − 0.384·61-s + 0.755·63-s − 0.868·65-s − 0.122·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59721984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59721984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 7 T + 37 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 12 T + 89 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 71 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T + 85 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 10 T + 114 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 15 T + 131 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 21 T + 227 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 3 T + 113 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T + 103 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 11 T + 171 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 163 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 21 T + 287 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−7.68813952936317797193188727882, −7.49174185690278150122121328506, −7.22325034740626506864746270079, −6.81108386078542963819171176447, −6.27717675271568572783353313241, −5.98853239954013305748434649626, −5.38709796497648731496636267061, −5.33461640311243278041538552483, −4.79862542729751992382232222643, −4.49096009984607168165422525718, −4.14736443020660706056601097468, −3.61083227672594259200323577511, −3.32947817317848907269650949570, −2.88978472574462379918799319639, −2.21428253813240829996975073078, −2.18739715156204261957333036861, −1.62282548934540466802950009127, −1.43526781970368412770385030881, 0, 0,
1.43526781970368412770385030881, 1.62282548934540466802950009127, 2.18739715156204261957333036861, 2.21428253813240829996975073078, 2.88978472574462379918799319639, 3.32947817317848907269650949570, 3.61083227672594259200323577511, 4.14736443020660706056601097468, 4.49096009984607168165422525718, 4.79862542729751992382232222643, 5.33461640311243278041538552483, 5.38709796497648731496636267061, 5.98853239954013305748434649626, 6.27717675271568572783353313241, 6.81108386078542963819171176447, 7.22325034740626506864746270079, 7.49174185690278150122121328506, 7.68813952936317797193188727882
