| L(s) = 1 | − 2·5-s + 2·11-s + 4·13-s − 4·17-s + 3·25-s − 4·29-s − 4·37-s − 12·41-s − 8·43-s − 6·49-s − 12·53-s − 4·55-s − 8·59-s − 4·61-s − 8·65-s + 8·67-s + 12·73-s + 8·79-s + 16·83-s + 8·85-s − 4·89-s − 12·97-s − 4·101-s + 16·103-s − 4·109-s − 20·113-s + 3·121-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.603·11-s + 1.10·13-s − 0.970·17-s + 3/5·25-s − 0.742·29-s − 0.657·37-s − 1.87·41-s − 1.21·43-s − 6/7·49-s − 1.64·53-s − 0.539·55-s − 1.04·59-s − 0.512·61-s − 0.992·65-s + 0.977·67-s + 1.40·73-s + 0.900·79-s + 1.75·83-s + 0.867·85-s − 0.423·89-s − 1.21·97-s − 0.398·101-s + 1.57·103-s − 0.383·109-s − 1.88·113-s + 3/11·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_4$ | \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 94 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 118 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 174 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 222 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 198 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−7.67286894829059156625538837961, −7.40871048849022923467788444950, −6.82353055819845980488646191408, −6.64523568710319627213938996293, −6.32589609665028194211938272550, −6.23715513788936876268844586197, −5.40666751161788063915724850754, −5.25407004494134010043449619331, −4.67786291770454480142662317627, −4.65922733166980815692968855512, −3.90645560350845354437640713767, −3.75207649237520966320957960969, −3.29872758565328995546213028662, −3.23135405032506434529958773422, −2.37304628919981464986767094000, −1.98973573013663824605392678924, −1.43573356079406208725427851611, −1.12204291446713510649357207894, 0, 0,
1.12204291446713510649357207894, 1.43573356079406208725427851611, 1.98973573013663824605392678924, 2.37304628919981464986767094000, 3.23135405032506434529958773422, 3.29872758565328995546213028662, 3.75207649237520966320957960969, 3.90645560350845354437640713767, 4.65922733166980815692968855512, 4.67786291770454480142662317627, 5.25407004494134010043449619331, 5.40666751161788063915724850754, 6.23715513788936876268844586197, 6.32589609665028194211938272550, 6.64523568710319627213938996293, 6.82353055819845980488646191408, 7.40871048849022923467788444950, 7.67286894829059156625538837961
