| L(s) = 1 | + 2-s + 4-s + 4·7-s + 8-s − 2·9-s + 4·14-s + 16-s − 2·18-s + 2·25-s + 4·28-s − 8·31-s + 32-s − 2·36-s + 12·41-s − 2·49-s + 2·50-s + 4·56-s − 8·62-s − 8·63-s + 64-s − 2·72-s − 8·73-s + 4·79-s − 5·81-s + 12·82-s − 12·89-s − 8·97-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.51·7-s + 0.353·8-s − 2/3·9-s + 1.06·14-s + 1/4·16-s − 0.471·18-s + 2/5·25-s + 0.755·28-s − 1.43·31-s + 0.176·32-s − 1/3·36-s + 1.87·41-s − 2/7·49-s + 0.282·50-s + 0.534·56-s − 1.01·62-s − 1.00·63-s + 1/8·64-s − 0.235·72-s − 0.936·73-s + 0.450·79-s − 5/9·81-s + 1.32·82-s − 1.27·89-s − 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.194848086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.194848086\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−10.65574890714556758651508610973, −9.859359817626190714145928206498, −9.241675483962209372351439072440, −8.705009097686523201686883341643, −8.152046643087779299299639815386, −7.70453567926153233157182003025, −7.18062082656271903203663127594, −6.48270922492735280387018630494, −5.68274234738021378017249582457, −5.42301847053767145238489399285, −4.65715382336057347013915350481, −4.19317114489120115070438582570, −3.28004535932058040813200560964, −2.43958508845422320755675728788, −1.51811583984037933887246360582,
1.51811583984037933887246360582, 2.43958508845422320755675728788, 3.28004535932058040813200560964, 4.19317114489120115070438582570, 4.65715382336057347013915350481, 5.42301847053767145238489399285, 5.68274234738021378017249582457, 6.48270922492735280387018630494, 7.18062082656271903203663127594, 7.70453567926153233157182003025, 8.152046643087779299299639815386, 8.705009097686523201686883341643, 9.241675483962209372351439072440, 9.859359817626190714145928206498, 10.65574890714556758651508610973
