| L(s) = 1 | − 2·3-s + 2·5-s − 2·7-s + 3·9-s + 8·11-s − 4·15-s − 10·17-s + 10·19-s + 4·21-s − 2·23-s − 2·25-s − 4·27-s + 4·31-s − 16·33-s − 4·35-s − 4·41-s + 2·43-s + 6·45-s + 8·47-s − 6·49-s + 20·51-s + 6·53-s + 16·55-s − 20·57-s + 8·59-s − 6·63-s + 6·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 0.755·7-s + 9-s + 2.41·11-s − 1.03·15-s − 2.42·17-s + 2.29·19-s + 0.872·21-s − 0.417·23-s − 2/5·25-s − 0.769·27-s + 0.718·31-s − 2.78·33-s − 0.676·35-s − 0.624·41-s + 0.304·43-s + 0.894·45-s + 1.16·47-s − 6/7·49-s + 2.80·51-s + 0.824·53-s + 2.15·55-s − 2.64·57-s + 1.04·59-s − 0.755·63-s + 0.733·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19501056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19501056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.647788908\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.647788908\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 10 T + 54 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 138 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 122 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 174 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 190 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.507716418999817764574548290266, −8.441408282366711073995019785350, −7.46528762319905767988794167842, −7.40494721912235168355962595950, −6.75262703450261873663320916954, −6.74698706265037305372256343962, −6.28434014368888752179255233395, −6.18206608454870779948807654788, −5.57231049812406729684594275727, −5.45658927915618676565695952104, −4.72794266175950756363362897090, −4.53991319426220476838226849336, −4.06345242718058504815478985112, −3.58506089018139398996366427890, −3.36601026636722088450397073921, −2.55101066063760646660556144537, −1.98720633005334702452871899136, −1.73023988982698844854951473365, −0.898696741706639839845683108283, −0.63678739587641530320474763007,
0.63678739587641530320474763007, 0.898696741706639839845683108283, 1.73023988982698844854951473365, 1.98720633005334702452871899136, 2.55101066063760646660556144537, 3.36601026636722088450397073921, 3.58506089018139398996366427890, 4.06345242718058504815478985112, 4.53991319426220476838226849336, 4.72794266175950756363362897090, 5.45658927915618676565695952104, 5.57231049812406729684594275727, 6.18206608454870779948807654788, 6.28434014368888752179255233395, 6.74698706265037305372256343962, 6.75262703450261873663320916954, 7.40494721912235168355962595950, 7.46528762319905767988794167842, 8.441408282366711073995019785350, 8.507716418999817764574548290266
