| L(s) = 1 | + 5-s + 2·7-s + 6·13-s − 2·17-s + 6·19-s − 6·25-s + 5·31-s + 2·35-s + 2·37-s + 3·41-s + 11·43-s − 2·47-s + 3·49-s + 11·53-s − 4·59-s − 5·61-s + 6·65-s − 7·67-s − 6·71-s − 5·73-s + 18·79-s − 12·83-s − 2·85-s + 26·89-s + 12·91-s + 6·95-s − 9·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s + 1.66·13-s − 0.485·17-s + 1.37·19-s − 6/5·25-s + 0.898·31-s + 0.338·35-s + 0.328·37-s + 0.468·41-s + 1.67·43-s − 0.291·47-s + 3/7·49-s + 1.51·53-s − 0.520·59-s − 0.640·61-s + 0.744·65-s − 0.855·67-s − 0.712·71-s − 0.585·73-s + 2.02·79-s − 1.31·83-s − 0.216·85-s + 2.75·89-s + 1.25·91-s + 0.615·95-s − 0.913·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18352656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18352656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.619617716\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.619617716\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - T + 7 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 5 T + 65 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 62 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 55 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 11 T + 113 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 82 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 11 T + 133 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 99 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 7 T + 143 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 5 T + 123 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 18 T + 226 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 26 T + 334 T^{2} - 26 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 9 T + 211 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
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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
−8.413667052895231905269585319245, −8.312221170000839073251357970616, −7.75151030425490729981516164182, −7.65705935279234566340282350770, −7.10389694933936418954119087365, −6.80071084547504318702766169108, −6.13841348612126649168166510717, −6.07385064255685252344337114756, −5.58713381178498715273151011829, −5.46195039768466721318032710249, −4.74240679608196964719085534582, −4.45682831339284094802878470208, −3.99365513929709601846282306303, −3.74125701247911617777978494011, −2.92800713428309059345534433600, −2.90743018769467865146798896447, −1.99354253016083482081161234246, −1.79095117409260482289328532715, −1.07253998566146547319456218513, −0.70475026488933325424144263335,
0.70475026488933325424144263335, 1.07253998566146547319456218513, 1.79095117409260482289328532715, 1.99354253016083482081161234246, 2.90743018769467865146798896447, 2.92800713428309059345534433600, 3.74125701247911617777978494011, 3.99365513929709601846282306303, 4.45682831339284094802878470208, 4.74240679608196964719085534582, 5.46195039768466721318032710249, 5.58713381178498715273151011829, 6.07385064255685252344337114756, 6.13841348612126649168166510717, 6.80071084547504318702766169108, 7.10389694933936418954119087365, 7.65705935279234566340282350770, 7.75151030425490729981516164182, 8.312221170000839073251357970616, 8.413667052895231905269585319245
