| L(s) = 1 | + 3-s + 2·5-s + 3·9-s + 7·11-s − 3·13-s + 2·15-s − 5·17-s − 2·19-s + 2·23-s + 3·25-s + 8·27-s − 3·29-s + 16·31-s + 7·33-s − 4·37-s − 3·39-s − 2·41-s − 6·43-s + 6·45-s − 3·47-s − 5·51-s + 10·53-s + 14·55-s − 2·57-s − 16·59-s − 6·61-s − 6·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 9-s + 2.11·11-s − 0.832·13-s + 0.516·15-s − 1.21·17-s − 0.458·19-s + 0.417·23-s + 3/5·25-s + 1.53·27-s − 0.557·29-s + 2.87·31-s + 1.21·33-s − 0.657·37-s − 0.480·39-s − 0.312·41-s − 0.914·43-s + 0.894·45-s − 0.437·47-s − 0.700·51-s + 1.37·53-s + 1.88·55-s − 0.264·57-s − 2.08·59-s − 0.768·61-s − 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.706781412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.706781412\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 32 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 50 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 62 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 10 T + 98 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 13 T + 126 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 226 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 9 T + 8 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
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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
−9.488991419004228700428521171191, −8.970413356972052551375878595018, −8.595084631129265987372967131490, −8.466197216153443266395986906478, −7.87275172656082828697953034656, −7.29225899201460954213517827007, −6.84341896572567532660166615993, −6.66863998041645263956395365037, −6.38759249505898261552433991224, −6.01971194581359503799376788947, −5.24403265531364040439452053680, −4.72533348316685646795190067079, −4.50448923309360709457113302045, −4.17829525733623613049379047106, −3.44813511706461029467473953842, −3.06967483490841699938799059773, −2.39889692749422277128051467732, −1.95062111430707587872792432362, −1.44858291980842852550362272970, −0.800343423597108698838844792882,
0.800343423597108698838844792882, 1.44858291980842852550362272970, 1.95062111430707587872792432362, 2.39889692749422277128051467732, 3.06967483490841699938799059773, 3.44813511706461029467473953842, 4.17829525733623613049379047106, 4.50448923309360709457113302045, 4.72533348316685646795190067079, 5.24403265531364040439452053680, 6.01971194581359503799376788947, 6.38759249505898261552433991224, 6.66863998041645263956395365037, 6.84341896572567532660166615993, 7.29225899201460954213517827007, 7.87275172656082828697953034656, 8.466197216153443266395986906478, 8.595084631129265987372967131490, 8.970413356972052551375878595018, 9.488991419004228700428521171191
