| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s − 4·6-s + 4·8-s + 3·9-s − 6·12-s − 12·13-s + 5·16-s − 2·17-s + 6·18-s + 4·19-s − 8·23-s − 8·24-s − 24·26-s − 4·27-s − 4·29-s − 8·31-s + 6·32-s − 4·34-s + 9·36-s − 4·37-s + 8·38-s + 24·39-s − 12·41-s − 4·43-s − 16·46-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 − 1.73·12-s − 3.32·13-s + 5/4·16-s − 0.485·17-s + 1.41·18-s + 0.917·19-s − 1.66·23-s − 1.63·24-s − 4.70·26-s − 0.769·27-s − 0.742·29-s − 1.43·31-s + 1.06·32-s − 0.685·34-s + 3/2·36-s − 0.657·37-s + 1.29·38-s + 3.84·39-s − 1.87·41-s − 0.609·43-s − 2.35·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 56 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 72 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 72 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 104 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 136 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 20 T + 222 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 168 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 146 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 16 T + 146 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−8.580146443321086626411745454568, −8.206315397634574447902406415734, −7.45397257341482684004048396476, −7.37949091513163869652225485929, −7.24214182222067515602782235079, −6.69184351502152784990078135798, −6.36345867043676123948464107705, −5.78841642242224113593090383967, −5.32107224286238112333698688631, −5.30243407290834490264870357348, −4.83414082823285989631889017844, −4.55708370717111953737147648800, −3.80873989216357019006401529082, −3.79062353951754526879715451853, −2.81354291612170889049388687258, −2.60833362751373309240544519149, −1.79684396230397726346931956942, −1.66325318979744769235207780364, 0, 0,
1.66325318979744769235207780364, 1.79684396230397726346931956942, 2.60833362751373309240544519149, 2.81354291612170889049388687258, 3.79062353951754526879715451853, 3.80873989216357019006401529082, 4.55708370717111953737147648800, 4.83414082823285989631889017844, 5.30243407290834490264870357348, 5.32107224286238112333698688631, 5.78841642242224113593090383967, 6.36345867043676123948464107705, 6.69184351502152784990078135798, 7.24214182222067515602782235079, 7.37949091513163869652225485929, 7.45397257341482684004048396476, 8.206315397634574447902406415734, 8.580146443321086626411745454568
