| L(s) = 1 | − 2·2-s + 2·4-s − 4·11-s − 10·13-s − 4·16-s + 8·22-s + 12·23-s + 25-s + 20·26-s + 8·32-s + 10·37-s − 8·44-s − 24·46-s + 8·47-s − 5·49-s − 2·50-s − 20·52-s − 16·59-s + 14·61-s − 8·64-s + 4·71-s − 10·73-s − 20·74-s + 12·83-s + 24·92-s − 16·94-s − 26·97-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 1.20·11-s − 2.77·13-s − 16-s + 1.70·22-s + 2.50·23-s + 1/5·25-s + 3.92·26-s + 1.41·32-s + 1.64·37-s − 1.20·44-s − 3.53·46-s + 1.16·47-s − 5/7·49-s − 0.282·50-s − 2.77·52-s − 2.08·59-s + 1.79·61-s − 64-s + 0.474·71-s − 1.17·73-s − 2.32·74-s + 1.31·83-s + 2.50·92-s − 1.65·94-s − 2.63·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 291600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 291600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
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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
−8.705039129288459634179104137275, −8.033038279448884875142132147382, −7.72379481902030800570658235358, −7.44584033135863585609716971257, −6.88066201922308126532422526866, −6.68799725367635564271959497809, −5.60485752225476628583929294992, −5.21952785456145416033046309322, −4.69221690938344723805408149899, −4.39646975608954166240974959735, −3.07119386896251421193504749200, −2.65254993398592503147695516254, −2.20929616241085403285528735477, −1.02045420469944533258202986585, 0,
1.02045420469944533258202986585, 2.20929616241085403285528735477, 2.65254993398592503147695516254, 3.07119386896251421193504749200, 4.39646975608954166240974959735, 4.69221690938344723805408149899, 5.21952785456145416033046309322, 5.60485752225476628583929294992, 6.68799725367635564271959497809, 6.88066201922308126532422526866, 7.44584033135863585609716971257, 7.72379481902030800570658235358, 8.033038279448884875142132147382, 8.705039129288459634179104137275
