| L(s) = 1 | + 4·2-s + 8·4-s − 2·7-s + 8·8-s − 2·11-s − 8·14-s − 4·16-s − 8·22-s + 2·23-s − 9·25-s − 16·28-s − 32·32-s + 6·37-s − 12·43-s − 16·44-s + 8·46-s − 3·49-s − 36·50-s + 12·53-s − 16·56-s − 64·64-s − 14·67-s + 6·71-s + 24·74-s + 4·77-s − 20·79-s − 48·86-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 4·4-s − 0.755·7-s + 2.82·8-s − 0.603·11-s − 2.13·14-s − 16-s − 1.70·22-s + 0.417·23-s − 9/5·25-s − 3.02·28-s − 5.65·32-s + 0.986·37-s − 1.82·43-s − 2.41·44-s + 1.17·46-s − 3/7·49-s − 5.09·50-s + 1.64·53-s − 2.13·56-s − 8·64-s − 1.71·67-s + 0.712·71-s + 2.78·74-s + 0.455·77-s − 2.25·79-s − 5.17·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480249 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480249 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−8.228854358433494441874279371750, −7.59708149153402218158879958117, −7.03477908276603020210779304212, −6.66503987304844816686807720890, −6.21101030419725238522849385481, −5.66407660024727617127120629681, −5.55294080899685406164716506758, −4.97685157209324421449824776256, −4.28111751945710160622264923994, −4.19085115174357929506261869806, −3.43654811333543968800336820680, −3.09000916592887094247460325417, −2.58331042621701060186581520596, −1.87316747898306017870215940641, 0,
1.87316747898306017870215940641, 2.58331042621701060186581520596, 3.09000916592887094247460325417, 3.43654811333543968800336820680, 4.19085115174357929506261869806, 4.28111751945710160622264923994, 4.97685157209324421449824776256, 5.55294080899685406164716506758, 5.66407660024727617127120629681, 6.21101030419725238522849385481, 6.66503987304844816686807720890, 7.03477908276603020210779304212, 7.59708149153402218158879958117, 8.228854358433494441874279371750
