| L(s) = 1 | + 2·3-s + 2·5-s − 2·7-s − 3·9-s − 2·11-s + 4·15-s + 4·17-s − 2·19-s − 4·21-s + 2·23-s + 3·25-s − 14·27-s − 2·29-s + 4·31-s − 4·33-s − 4·35-s + 2·37-s − 2·41-s − 10·43-s − 6·45-s − 2·47-s − 8·49-s + 8·51-s + 8·53-s − 4·55-s − 4·57-s + 2·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s − 0.755·7-s − 9-s − 0.603·11-s + 1.03·15-s + 0.970·17-s − 0.458·19-s − 0.872·21-s + 0.417·23-s + 3/5·25-s − 2.69·27-s − 0.371·29-s + 0.718·31-s − 0.696·33-s − 0.676·35-s + 0.328·37-s − 0.312·41-s − 1.52·43-s − 0.894·45-s − 0.291·47-s − 8/7·49-s + 1.12·51-s + 1.09·53-s − 0.539·55-s − 0.529·57-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 11 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 63 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 72 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 71 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 108 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 110 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T + 48 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 68 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 28 T + 339 T^{2} + 28 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 14 T + 132 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 184 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 120 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−7.63818020200118032226837548279, −7.61085124528429823289127413838, −6.92190055539164158013273450935, −6.87073591126437149246899614785, −6.15641909326165511211523047866, −5.97462129512983394624851581116, −5.72861389688663613063425552113, −5.41432155733738407711773379750, −4.79896205586230872458264686628, −4.72255505714317392565588587346, −3.81873075310955507463933280908, −3.71940610262119284610931353134, −3.06535066230145378492934383481, −2.94867916833247986265263684594, −2.60734540209516801941439535382, −2.28975377025911714998065211130, −1.50295340077652371938691194113, −1.32588275026516163104589281609, 0, 0,
1.32588275026516163104589281609, 1.50295340077652371938691194113, 2.28975377025911714998065211130, 2.60734540209516801941439535382, 2.94867916833247986265263684594, 3.06535066230145378492934383481, 3.71940610262119284610931353134, 3.81873075310955507463933280908, 4.72255505714317392565588587346, 4.79896205586230872458264686628, 5.41432155733738407711773379750, 5.72861389688663613063425552113, 5.97462129512983394624851581116, 6.15641909326165511211523047866, 6.87073591126437149246899614785, 6.92190055539164158013273450935, 7.61085124528429823289127413838, 7.63818020200118032226837548279
