| L(s) = 1 | + 3·2-s + 3·3-s + 8·4-s − 3·5-s + 9·6-s + 45·8-s − 9·10-s + 15·11-s + 24·12-s + 128·13-s − 9·15-s + 135·16-s + 84·17-s − 16·19-s − 24·20-s + 45·22-s + 84·23-s + 135·24-s + 125·25-s + 384·26-s − 27·27-s − 594·29-s − 27·30-s − 253·31-s + 360·32-s + 45·33-s + 252·34-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 0.577·3-s + 4-s − 0.268·5-s + 0.612·6-s + 1.98·8-s − 0.284·10-s + 0.411·11-s + 0.577·12-s + 2.73·13-s − 0.154·15-s + 2.10·16-s + 1.19·17-s − 0.193·19-s − 0.268·20-s + 0.436·22-s + 0.761·23-s + 1.14·24-s + 25-s + 2.89·26-s − 0.192·27-s − 3.80·29-s − 0.164·30-s − 1.46·31-s + 1.98·32-s + 0.237·33-s + 1.27·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.071238859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.071238859\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T - 116 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 15 T - 1106 T^{2} - 15 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 64 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 84 T + 2143 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 16 T - 6603 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 84 T - 5111 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 297 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 253 T + 34218 T^{2} + 253 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 316 T + 49203 T^{2} - 316 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 360 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 26 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 30 T - 102923 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 363 T - 17108 T^{2} + 363 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 15 T - 205154 T^{2} + 15 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 118 T - 213057 T^{2} + 118 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 370 T - 163863 T^{2} - 370 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 342 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 362 T - 257973 T^{2} - 362 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 467 T - 274950 T^{2} + 467 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 477 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 906 T + 115867 T^{2} - 906 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 503 T + p^{3} T^{2} )^{2} \) |
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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
−13.05253585052028495242296659509, −12.70896822669867857549390891060, −11.71474196055688402396418224015, −11.31591211882210635298441118862, −10.89723522390686879738388645702, −10.72163773940034770007008903459, −9.764356135350725708438517771339, −9.210009467697763846336108466812, −8.505395791247484752971236149756, −8.131431321827146537120156053627, −7.33475489140444438216219836936, −7.12422991627852628487606547170, −6.14557377251303475641347644439, −5.70546428097437895329684621541, −5.06508962445991743858441987979, −4.10924363295733866121086820539, −3.49140915880193765976326389736, −3.40932366894813089321204557860, −1.78412743762629322276598970783, −1.32128003035094388887531434919,
1.32128003035094388887531434919, 1.78412743762629322276598970783, 3.40932366894813089321204557860, 3.49140915880193765976326389736, 4.10924363295733866121086820539, 5.06508962445991743858441987979, 5.70546428097437895329684621541, 6.14557377251303475641347644439, 7.12422991627852628487606547170, 7.33475489140444438216219836936, 8.131431321827146537120156053627, 8.505395791247484752971236149756, 9.210009467697763846336108466812, 9.764356135350725708438517771339, 10.72163773940034770007008903459, 10.89723522390686879738388645702, 11.31591211882210635298441118862, 11.71474196055688402396418224015, 12.70896822669867857549390891060, 13.05253585052028495242296659509
