L(s) = 1 | − 2·2-s + 3·4-s + 2·7-s − 4·8-s − 4·11-s − 2·13-s − 4·14-s + 5·16-s + 10·17-s − 8·19-s + 8·22-s + 6·23-s + 4·26-s + 6·28-s − 2·29-s + 2·31-s − 6·32-s − 20·34-s − 12·37-s + 16·38-s + 8·41-s − 14·43-s − 12·44-s − 12·46-s − 4·47-s + 3·49-s − 6·52-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.755·7-s − 1.41·8-s − 1.20·11-s − 0.554·13-s − 1.06·14-s + 5/4·16-s + 2.42·17-s − 1.83·19-s + 1.70·22-s + 1.25·23-s + 0.784·26-s + 1.13·28-s − 0.371·29-s + 0.359·31-s − 1.06·32-s − 3.42·34-s − 1.97·37-s + 2.59·38-s + 1.24·41-s − 2.13·43-s − 1.80·44-s − 1.76·46-s − 0.583·47-s + 3/7·49-s − 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89302500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89302500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 11 | $D_{4}$ | \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 17 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 44 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 45 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 49 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 100 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 86 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 18 T + 175 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 61 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 20 T + 248 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 130 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 219 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 188 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.52169735038526594426839758358, −7.48302726798195083966999446695, −6.88322974433633166573183490075, −6.80229851079147764778962192005, −6.06983343806286980930378485760, −6.06447755394058654626425071153, −5.42153427530999723581217965061, −5.17794213574374964783120977722, −4.85315816106264572028248215107, −4.63132816824638711469590151465, −3.72783940256780572987684723712, −3.59260068928039736941010319144, −3.10869790912962184322525117511, −2.61756827609882675401432251485, −2.36023941961545774260371298498, −1.81766535566498104078896953158, −1.33627175476805795445693567454, −1.06820064405993836423504438222, 0, 0,
1.06820064405993836423504438222, 1.33627175476805795445693567454, 1.81766535566498104078896953158, 2.36023941961545774260371298498, 2.61756827609882675401432251485, 3.10869790912962184322525117511, 3.59260068928039736941010319144, 3.72783940256780572987684723712, 4.63132816824638711469590151465, 4.85315816106264572028248215107, 5.17794213574374964783120977722, 5.42153427530999723581217965061, 6.06447755394058654626425071153, 6.06983343806286980930378485760, 6.80229851079147764778962192005, 6.88322974433633166573183490075, 7.48302726798195083966999446695, 7.52169735038526594426839758358