| L(s) = 1 | − 5·2-s + 7·3-s + 8·4-s + 14·5-s − 35·6-s − 13·7-s + 5·8-s + 27·9-s − 70·10-s − 26·11-s + 56·12-s + 65·14-s + 98·15-s − 25·16-s − 77·17-s − 135·18-s − 126·19-s + 112·20-s − 91·21-s + 130·22-s + 96·23-s + 35·24-s − 103·25-s + 224·27-s − 104·28-s + 82·29-s − 490·30-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 1.34·3-s + 4-s + 1.25·5-s − 2.38·6-s − 0.701·7-s + 0.220·8-s + 9-s − 2.21·10-s − 0.712·11-s + 1.34·12-s + 1.24·14-s + 1.68·15-s − 0.390·16-s − 1.09·17-s − 1.76·18-s − 1.52·19-s + 1.25·20-s − 0.945·21-s + 1.25·22-s + 0.870·23-s + 0.297·24-s − 0.823·25-s + 1.59·27-s − 0.701·28-s + 0.525·29-s − 2.98·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28561 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28561 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8750441717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8750441717\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 5 T + 17 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 7 T + 22 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 7 T + p^{3} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 13 T - 174 T^{2} + 13 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 26 T - 655 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 77 T + 1016 T^{2} + 77 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 126 T + 9017 T^{2} + 126 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 96 T - 2951 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 82 T - 17665 T^{2} - 82 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 196 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 131 T - 33492 T^{2} + 131 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 336 T + 43975 T^{2} - 336 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 201 T - 39106 T^{2} - 201 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 105 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 432 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 294 T - 118943 T^{2} + 294 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 56 T - 223845 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 478 T - 72279 T^{2} - 478 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 9 T - 357830 T^{2} - 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 98 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 1304 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 308 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 1190 T + 711131 T^{2} + 1190 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 70 T - 907773 T^{2} - 70 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−12.65952483754264354068649741060, −12.36032747335644965472207725732, −10.99918459226669365897682717624, −10.75639291940734256476841036414, −10.41953862586082272335670702926, −9.577893585456094749499755765687, −9.534210322362023116304212955912, −8.960116505072500222169624362845, −8.852583368604701797208425053984, −8.112211098299518116727171970876, −7.74579401140566999044386890038, −7.02486520333471652254689326860, −6.40532387260018605088522162402, −5.82733478085236609519846986499, −4.89114988219898330083302468811, −4.05678550095286923459644667792, −3.12907310242796333162048933211, −2.26740113364697859585444596896, −1.90016243719302224106282098731, −0.51709899821583641084620976354,
0.51709899821583641084620976354, 1.90016243719302224106282098731, 2.26740113364697859585444596896, 3.12907310242796333162048933211, 4.05678550095286923459644667792, 4.89114988219898330083302468811, 5.82733478085236609519846986499, 6.40532387260018605088522162402, 7.02486520333471652254689326860, 7.74579401140566999044386890038, 8.112211098299518116727171970876, 8.852583368604701797208425053984, 8.960116505072500222169624362845, 9.534210322362023116304212955912, 9.577893585456094749499755765687, 10.41953862586082272335670702926, 10.75639291940734256476841036414, 10.99918459226669365897682717624, 12.36032747335644965472207725732, 12.65952483754264354068649741060
