| L(s) = 1 | − 2-s − 2·3-s − 2·4-s − 2·5-s + 2·6-s − 2·7-s + 3·8-s + 3·9-s + 2·10-s + 4·12-s − 6·13-s + 2·14-s + 4·15-s + 16-s + 2·17-s − 3·18-s − 4·19-s + 4·20-s + 4·21-s + 4·23-s − 6·24-s + 3·25-s + 6·26-s − 4·27-s + 4·28-s − 2·29-s − 4·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 4-s − 0.894·5-s + 0.816·6-s − 0.755·7-s + 1.06·8-s + 9-s + 0.632·10-s + 1.15·12-s − 1.66·13-s + 0.534·14-s + 1.03·15-s + 1/4·16-s + 0.485·17-s − 0.707·18-s − 0.917·19-s + 0.894·20-s + 0.872·21-s + 0.834·23-s − 1.22·24-s + 3/5·25-s + 1.17·26-s − 0.769·27-s + 0.755·28-s − 0.371·29-s − 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3144182041\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3144182041\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 54 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 + 16 T + 126 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 90 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 53 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 98 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 111 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 18 T + 210 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 20 T + 222 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 162 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 149 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 14 T + 222 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 30 T + 414 T^{2} - 30 p T^{3} + p^{2} T^{4} \) |
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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
−9.427735204239699058008387852557, −9.233793253763432116600506287537, −8.533172214132035740068902588263, −8.527456742297054131306434749811, −7.78746572535323430950112224944, −7.58416251207348191836523596488, −7.11851809131232780418872981809, −6.78981741597787002403718131194, −6.34515545145239911584186514410, −5.86116861770557489530215626444, −5.27108553574512450169205395726, −4.91084830678310215610129275138, −4.67953276280956347706701053403, −4.21676799355029820028982200029, −3.65502474291857546106345561192, −3.24739389726273723053510899496, −2.49151405356455115431622926798, −1.78017279561078941584900007992, −0.66467287144476195552021162034, −0.45256107468636918978321523588,
0.45256107468636918978321523588, 0.66467287144476195552021162034, 1.78017279561078941584900007992, 2.49151405356455115431622926798, 3.24739389726273723053510899496, 3.65502474291857546106345561192, 4.21676799355029820028982200029, 4.67953276280956347706701053403, 4.91084830678310215610129275138, 5.27108553574512450169205395726, 5.86116861770557489530215626444, 6.34515545145239911584186514410, 6.78981741597787002403718131194, 7.11851809131232780418872981809, 7.58416251207348191836523596488, 7.78746572535323430950112224944, 8.527456742297054131306434749811, 8.533172214132035740068902588263, 9.233793253763432116600506287537, 9.427735204239699058008387852557
