| L(s) = 1 | + 2-s + 4-s + 7·7-s + 8-s + 7·14-s + 16-s + 3·23-s − 25-s + 7·28-s − 8·31-s + 32-s − 6·41-s + 3·46-s − 21·47-s + 25·49-s − 50-s + 7·56-s − 8·62-s + 64-s − 9·71-s − 14·73-s + 7·79-s − 6·82-s − 9·89-s + 3·92-s − 21·94-s − 2·97-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 2.64·7-s + 0.353·8-s + 1.87·14-s + 1/4·16-s + 0.625·23-s − 1/5·25-s + 1.32·28-s − 1.43·31-s + 0.176·32-s − 0.937·41-s + 0.442·46-s − 3.06·47-s + 25/7·49-s − 0.141·50-s + 0.935·56-s − 1.01·62-s + 1/8·64-s − 1.06·71-s − 1.63·73-s + 0.787·79-s − 0.662·82-s − 0.953·89-s + 0.312·92-s − 2.16·94-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.127265868\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.127265868\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
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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
−9.775397228773314031548258520109, −8.849278284468999033112715728732, −8.603731945338581425374364462265, −8.081259433067641899978730792757, −7.58276308738234069759401620165, −7.25619069582781203896870502714, −6.54896665786841412417648128881, −5.83858853924817962467467515672, −5.23035883253216452834758051317, −4.89159680050254016542207058757, −4.50626253103602407159101930849, −3.73500150366978422036203499556, −2.96117450780556495599330740307, −1.83803415980181296688864265428, −1.57086113534161898750717261802,
1.57086113534161898750717261802, 1.83803415980181296688864265428, 2.96117450780556495599330740307, 3.73500150366978422036203499556, 4.50626253103602407159101930849, 4.89159680050254016542207058757, 5.23035883253216452834758051317, 5.83858853924817962467467515672, 6.54896665786841412417648128881, 7.25619069582781203896870502714, 7.58276308738234069759401620165, 8.081259433067641899978730792757, 8.603731945338581425374364462265, 8.849278284468999033112715728732, 9.775397228773314031548258520109
