| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s − 4·9-s + 5·16-s + 8·18-s − 5·25-s − 6·32-s − 12·36-s + 12·37-s − 7·49-s + 10·50-s + 7·64-s + 24·67-s + 16·72-s − 24·74-s − 16·79-s + 7·81-s + 14·98-s − 15·100-s + 4·121-s + 127-s − 8·128-s + 131-s − 48·134-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s − 4/3·9-s + 5/4·16-s + 1.88·18-s − 25-s − 1.06·32-s − 2·36-s + 1.97·37-s − 49-s + 1.41·50-s + 7/8·64-s + 2.93·67-s + 1.88·72-s − 2.78·74-s − 1.80·79-s + 7/9·81-s + 1.41·98-s − 3/2·100-s + 4/11·121-s + 0.0887·127-s − 0.707·128-s + 0.0873·131-s − 4.14·134-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| 13 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 100 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−8.131618642101943116882547708661, −7.72437894098703989541368188556, −7.32884906401321983921060616831, −6.72018335255476099587435107436, −6.25825842651084024221299042653, −5.97534927719124518967832195894, −5.44177758047740457664241158059, −4.95547336736508147411603800608, −4.13419533428884077603165499307, −3.60435308896677010671386138747, −2.87856638244476110261940156287, −2.51492111414859397487305004781, −1.84930248187087390627189901757, −0.932808401678431971346665144078, 0,
0.932808401678431971346665144078, 1.84930248187087390627189901757, 2.51492111414859397487305004781, 2.87856638244476110261940156287, 3.60435308896677010671386138747, 4.13419533428884077603165499307, 4.95547336736508147411603800608, 5.44177758047740457664241158059, 5.97534927719124518967832195894, 6.25825842651084024221299042653, 6.72018335255476099587435107436, 7.32884906401321983921060616831, 7.72437894098703989541368188556, 8.131618642101943116882547708661
