| L(s) = 1 | − 6·5-s − 4·7-s + 9-s − 2·13-s − 12·23-s + 17·25-s − 6·29-s + 24·35-s − 6·45-s − 2·49-s − 18·53-s − 12·59-s − 4·63-s + 12·65-s − 16·67-s − 8·81-s + 12·83-s + 8·91-s + 8·103-s − 36·107-s + 10·109-s + 72·115-s − 2·117-s + 17·121-s − 18·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 2.68·5-s − 1.51·7-s + 1/3·9-s − 0.554·13-s − 2.50·23-s + 17/5·25-s − 1.11·29-s + 4.05·35-s − 0.894·45-s − 2/7·49-s − 2.47·53-s − 1.56·59-s − 0.503·63-s + 1.48·65-s − 1.95·67-s − 8/9·81-s + 1.31·83-s + 0.838·91-s + 0.788·103-s − 3.48·107-s + 0.957·109-s + 6.71·115-s − 0.184·117-s + 1.54·121-s − 1.60·125-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 215296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 29 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 89 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 113 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
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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
−10.76037965363535269672893604370, −10.56451201665711882641045625237, −9.785131982878043230797457587065, −9.529681127614179617511111048807, −9.157887318006110674915384449183, −8.193804114258180897113054308921, −8.071792915173922449640174275938, −7.74540275600018431843168094756, −7.18503374698014605504712059025, −6.86198053722046681332203074081, −6.09672650055990879875333833246, −5.86388849947578803283468556484, −4.66724266733739728355968639383, −4.45244375101330238868170389449, −3.84793333678063425902457357771, −3.39254278311452190233534856751, −3.05816148004845947873979101806, −1.85157816470025325503719636321, 0, 0,
1.85157816470025325503719636321, 3.05816148004845947873979101806, 3.39254278311452190233534856751, 3.84793333678063425902457357771, 4.45244375101330238868170389449, 4.66724266733739728355968639383, 5.86388849947578803283468556484, 6.09672650055990879875333833246, 6.86198053722046681332203074081, 7.18503374698014605504712059025, 7.74540275600018431843168094756, 8.071792915173922449640174275938, 8.193804114258180897113054308921, 9.157887318006110674915384449183, 9.529681127614179617511111048807, 9.785131982878043230797457587065, 10.56451201665711882641045625237, 10.76037965363535269672893604370
