L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 2·6-s − 2·9-s − 2·11-s + 2·12-s + 8·13-s − 4·16-s − 4·18-s − 4·22-s + 2·23-s − 9·25-s + 16·26-s − 5·27-s − 8·32-s − 2·33-s − 4·36-s + 6·37-s + 8·39-s − 4·44-s + 4·46-s − 16·47-s − 4·48-s − 10·49-s − 18·50-s + 16·52-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s − 2/3·9-s − 0.603·11-s + 0.577·12-s + 2.21·13-s − 16-s − 0.942·18-s − 0.852·22-s + 0.417·23-s − 9/5·25-s + 3.13·26-s − 0.962·27-s − 1.41·32-s − 0.348·33-s − 2/3·36-s + 0.986·37-s + 1.28·39-s − 0.603·44-s + 0.589·46-s − 2.33·47-s − 0.577·48-s − 1.42·49-s − 2.54·50-s + 2.21·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.457371241\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.457371241\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−11.14136051051457299622132768480, −10.88023677523055899886301116177, −9.701226498296212058900989539920, −9.531323384279074079649336150802, −8.595888941067365837350463793393, −8.228854358433494441874279371750, −7.81893826163275369649161903506, −6.66503987304844816686807720890, −6.29323672924194667424631097798, −5.66407660024727617127120629681, −5.16796004190121525070885092557, −4.19085115174357929506261869806, −3.57632858228471856718550416679, −3.09000916592887094247460325417, −2.01168702564605014700712887669,
2.01168702564605014700712887669, 3.09000916592887094247460325417, 3.57632858228471856718550416679, 4.19085115174357929506261869806, 5.16796004190121525070885092557, 5.66407660024727617127120629681, 6.29323672924194667424631097798, 6.66503987304844816686807720890, 7.81893826163275369649161903506, 8.228854358433494441874279371750, 8.595888941067365837350463793393, 9.531323384279074079649336150802, 9.701226498296212058900989539920, 10.88023677523055899886301116177, 11.14136051051457299622132768480