| L(s) = 1 | + 2·3-s + 3·9-s − 12·17-s − 12·19-s − 10·25-s + 4·27-s − 12·41-s − 12·43-s − 12·49-s − 24·51-s − 24·57-s + 8·67-s − 20·75-s + 5·81-s + 24·83-s + 12·89-s − 24·97-s + 24·107-s − 12·113-s − 22·121-s − 24·123-s + 127-s − 24·129-s + 131-s + 137-s + 139-s − 24·147-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s − 2.91·17-s − 2.75·19-s − 2·25-s + 0.769·27-s − 1.87·41-s − 1.82·43-s − 1.71·49-s − 3.36·51-s − 3.17·57-s + 0.977·67-s − 2.30·75-s + 5/9·81-s + 2.63·83-s + 1.27·89-s − 2.43·97-s + 2.32·107-s − 1.12·113-s − 2·121-s − 2.16·123-s + 0.0887·127-s − 2.11·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.97·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9437184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9437184 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 156 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
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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
−8.454853781367797756636553630687, −8.363558190515066965444689061413, −7.80625480387228563532476004218, −7.64626842078064387369624448832, −6.74933700970922975173053735041, −6.70630120667193617066883545007, −6.33653705252243361476186371196, −6.27654155638568826457237679751, −5.16788100506058689876652617387, −5.06575504422163708506155976400, −4.36909394286918338504644771464, −4.28502186995316375831689342527, −3.64572946380750709897556219425, −3.54164608461105587331387980920, −2.58580379467367364315928798204, −2.32936025813766307004835379932, −1.85832827807362291286631062101, −1.68386498468344546732835314810, 0, 0,
1.68386498468344546732835314810, 1.85832827807362291286631062101, 2.32936025813766307004835379932, 2.58580379467367364315928798204, 3.54164608461105587331387980920, 3.64572946380750709897556219425, 4.28502186995316375831689342527, 4.36909394286918338504644771464, 5.06575504422163708506155976400, 5.16788100506058689876652617387, 6.27654155638568826457237679751, 6.33653705252243361476186371196, 6.70630120667193617066883545007, 6.74933700970922975173053735041, 7.64626842078064387369624448832, 7.80625480387228563532476004218, 8.363558190515066965444689061413, 8.454853781367797756636553630687
