| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s − 4·6-s + 4·8-s + 3·9-s − 2·11-s − 6·12-s − 2·13-s + 5·16-s − 2·17-s + 6·18-s − 6·19-s − 4·22-s − 4·23-s − 8·24-s − 8·25-s − 4·26-s − 4·27-s + 10·29-s − 8·31-s + 6·32-s + 4·33-s − 4·34-s + 9·36-s + 4·37-s − 12·38-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s − 1.63·6-s + 1.41·8-s + 9-s − 0.603·11-s − 1.73·12-s − 0.554·13-s + 5/4·16-s − 0.485·17-s + 1.41·18-s − 1.37·19-s − 0.852·22-s − 0.834·23-s − 1.63·24-s − 8/5·25-s − 0.784·26-s − 0.769·27-s + 1.85·29-s − 1.43·31-s + 1.06·32-s + 0.696·33-s − 0.685·34-s + 3/2·36-s + 0.657·37-s − 1.94·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 116 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 82 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 2 T - 21 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 18 T + 197 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T + 105 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 18 T + 183 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 132 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 164 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 100 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.031219295120492706746524736949, −7.84850906590055272423461774409, −7.41707988597355640892327764866, −7.12046997572306609777748996275, −6.47036919369068421616856970958, −6.40072757516301916714812063592, −5.95088384124794601677579232196, −5.88626898385872891247246338710, −5.13138844727790027209139287485, −4.90526130884327088092599661301, −4.56369999601239931200702352068, −4.37277715543162601329118615576, −3.66674595104966734801497342161, −3.46826117953080856358121733289, −2.74801650381568440893957762645, −2.33094316527121108743885927061, −1.76592773817637088990580844937, −1.45116518149866209303149119726, 0, 0,
1.45116518149866209303149119726, 1.76592773817637088990580844937, 2.33094316527121108743885927061, 2.74801650381568440893957762645, 3.46826117953080856358121733289, 3.66674595104966734801497342161, 4.37277715543162601329118615576, 4.56369999601239931200702352068, 4.90526130884327088092599661301, 5.13138844727790027209139287485, 5.88626898385872891247246338710, 5.95088384124794601677579232196, 6.40072757516301916714812063592, 6.47036919369068421616856970958, 7.12046997572306609777748996275, 7.41707988597355640892327764866, 7.84850906590055272423461774409, 8.031219295120492706746524736949
