| L(s) = 1 | − 2·2-s + 3·3-s + 3·4-s − 2·5-s − 6·6-s − 2·7-s − 4·8-s + 2·9-s + 4·10-s + 9·12-s − 4·13-s + 4·14-s − 6·15-s + 5·16-s + 17-s − 4·18-s − 7·19-s − 6·20-s − 6·21-s + 12·23-s − 12·24-s + 3·25-s + 8·26-s − 6·27-s − 6·28-s − 2·29-s + 12·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.73·3-s + 3/2·4-s − 0.894·5-s − 2.44·6-s − 0.755·7-s − 1.41·8-s + 2/3·9-s + 1.26·10-s + 2.59·12-s − 1.10·13-s + 1.06·14-s − 1.54·15-s + 5/4·16-s + 0.242·17-s − 0.942·18-s − 1.60·19-s − 1.34·20-s − 1.30·21-s + 2.50·23-s − 2.44·24-s + 3/5·25-s + 1.56·26-s − 1.15·27-s − 1.13·28-s − 0.371·29-s + 2.19·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - p T + 7 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - T + 33 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $C_4$ | \( 1 + 7 T + 39 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 54 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 5 T + 27 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3 T + 77 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 14 T + 138 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 13 T + 159 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_4$ | \( 1 - 16 T + 166 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - T + 103 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 106 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 21 T + 245 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 154 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 5 T + 111 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 15 T + 203 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - T + 93 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−7.57880156092898263352805140912, −7.46264429169674004532743692244, −7.21189559697520536471377097697, −6.87594465866567570213894041460, −6.34922902067241816381709942199, −6.25558135739230496028405748544, −5.42167959254139053064542883107, −5.38928741433899459529737440625, −4.67816911348486045534147491870, −4.30819419361542607104746686627, −3.71539330290363594427277486396, −3.66185719641731472701115974086, −2.90981253364544447084092289390, −2.84849362557972434686489115053, −2.38245051820441006457868111340, −2.34397077772204736041348114058, −1.37022022837728817710405307204, −1.02426882443860840444541055592, 0, 0,
1.02426882443860840444541055592, 1.37022022837728817710405307204, 2.34397077772204736041348114058, 2.38245051820441006457868111340, 2.84849362557972434686489115053, 2.90981253364544447084092289390, 3.66185719641731472701115974086, 3.71539330290363594427277486396, 4.30819419361542607104746686627, 4.67816911348486045534147491870, 5.38928741433899459529737440625, 5.42167959254139053064542883107, 6.25558135739230496028405748544, 6.34922902067241816381709942199, 6.87594465866567570213894041460, 7.21189559697520536471377097697, 7.46264429169674004532743692244, 7.57880156092898263352805140912
