| L(s) = 1 | − 3·5-s + 2·11-s + 4·13-s − 6·17-s − 6·19-s − 3·23-s + 25-s − 6·29-s + 15·31-s + 37-s − 6·43-s − 8·47-s − 14·49-s − 6·55-s − 13·59-s − 6·61-s − 12·65-s − 3·67-s + 19·71-s − 12·73-s + 18·79-s + 2·83-s + 18·85-s + 3·89-s + 18·95-s − 97-s − 18·101-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.603·11-s + 1.10·13-s − 1.45·17-s − 1.37·19-s − 0.625·23-s + 1/5·25-s − 1.11·29-s + 2.69·31-s + 0.164·37-s − 0.914·43-s − 1.16·47-s − 2·49-s − 0.809·55-s − 1.69·59-s − 0.768·61-s − 1.48·65-s − 0.366·67-s + 2.25·71-s − 1.40·73-s + 2.02·79-s + 0.219·83-s + 1.95·85-s + 0.317·89-s + 1.84·95-s − 0.101·97-s − 1.79·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10036224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10036224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 15 T + 114 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T + 36 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 13 T + 122 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T - 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 3 T + 98 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 19 T + 194 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 18 T + 222 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3 T + 176 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T + 156 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−8.270014063154563479463727586111, −8.245626168003335252540146068248, −7.76345369951209799085435051763, −7.67157290009868726354504921527, −6.69831114291708554928070266074, −6.59283119029711512261613926079, −6.34755789326942501570946477295, −6.19121182794177442485949515601, −5.28596208732442966384738442766, −4.94702257623900185278320827007, −4.37077969356154167704719650484, −4.27533101352841133875950963983, −3.69127476549459434347909838450, −3.59800863294767558936023093756, −2.80128735646112043636618196900, −2.41524788257273881335545028224, −1.62732435612824663355192277690, −1.29319010062930106062844847153, 0, 0,
1.29319010062930106062844847153, 1.62732435612824663355192277690, 2.41524788257273881335545028224, 2.80128735646112043636618196900, 3.59800863294767558936023093756, 3.69127476549459434347909838450, 4.27533101352841133875950963983, 4.37077969356154167704719650484, 4.94702257623900185278320827007, 5.28596208732442966384738442766, 6.19121182794177442485949515601, 6.34755789326942501570946477295, 6.59283119029711512261613926079, 6.69831114291708554928070266074, 7.67157290009868726354504921527, 7.76345369951209799085435051763, 8.245626168003335252540146068248, 8.270014063154563479463727586111
