| L(s) = 1 | + 2-s − 2·3-s + 4-s − 3·5-s − 2·6-s + 3·8-s + 3·9-s − 3·10-s + 11-s − 2·12-s + 6·15-s + 16-s + 2·17-s + 3·18-s − 3·19-s − 3·20-s + 22-s − 9·23-s − 6·24-s + 25-s − 4·27-s + 6·30-s + 2·31-s − 32-s − 2·33-s + 2·34-s + 3·36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 1.34·5-s − 0.816·6-s + 1.06·8-s + 9-s − 0.948·10-s + 0.301·11-s − 0.577·12-s + 1.54·15-s + 1/4·16-s + 0.485·17-s + 0.707·18-s − 0.688·19-s − 0.670·20-s + 0.213·22-s − 1.87·23-s − 1.22·24-s + 1/5·25-s − 0.769·27-s + 1.09·30-s + 0.359·31-s − 0.176·32-s − 0.348·33-s + 0.342·34-s + 1/2·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74287161 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74287161 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.220835199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.220835199\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 9 T + 62 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 14 T + 126 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 10 T + 130 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 14 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 174 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 170 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 226 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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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
−8.020757623601193075363433868366, −7.44213898285024669158691023291, −7.13953269882908177299210670535, −7.13250115659064670123311541214, −6.31610962444428892838095359546, −6.12653492793073990056231004124, −6.08754508359286840425909856975, −5.55861675642753164137030287040, −4.95087765292144465329289779295, −4.83902056289082627666391944339, −4.41962200006628570555699403821, −4.15746092784456538767064782331, −3.77963422100784221929763888969, −3.56218907638796309872847290657, −3.07310965062597667821386122448, −2.18199937180841988399862425610, −2.11356312485775401119011538964, −1.56327244911823841048866994562, −0.71335762196563185148936138320, −0.47876652526277408190701621166,
0.47876652526277408190701621166, 0.71335762196563185148936138320, 1.56327244911823841048866994562, 2.11356312485775401119011538964, 2.18199937180841988399862425610, 3.07310965062597667821386122448, 3.56218907638796309872847290657, 3.77963422100784221929763888969, 4.15746092784456538767064782331, 4.41962200006628570555699403821, 4.83902056289082627666391944339, 4.95087765292144465329289779295, 5.55861675642753164137030287040, 6.08754508359286840425909856975, 6.12653492793073990056231004124, 6.31610962444428892838095359546, 7.13250115659064670123311541214, 7.13953269882908177299210670535, 7.44213898285024669158691023291, 8.020757623601193075363433868366
