| L(s) = 1 | − 2·3-s + 4·7-s + 3·9-s + 8·13-s + 8·17-s + 4·19-s − 8·21-s − 2·23-s − 4·27-s + 4·29-s − 4·37-s − 16·39-s − 4·41-s − 4·43-s − 12·47-s + 8·49-s − 16·51-s − 8·53-s − 8·57-s − 4·59-s − 12·61-s + 12·63-s − 4·67-s + 4·69-s − 4·73-s + 20·79-s + 5·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.51·7-s + 9-s + 2.21·13-s + 1.94·17-s + 0.917·19-s − 1.74·21-s − 0.417·23-s − 0.769·27-s + 0.742·29-s − 0.657·37-s − 2.56·39-s − 0.624·41-s − 0.609·43-s − 1.75·47-s + 8/7·49-s − 2.24·51-s − 1.09·53-s − 1.05·57-s − 0.520·59-s − 1.53·61-s + 1.51·63-s − 0.488·67-s + 0.481·69-s − 0.468·73-s + 2.25·79-s + 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76176 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.446136669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.446136669\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 8 T + 40 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 32 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 80 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 82 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 128 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 20 T + 248 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 20 T + 254 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
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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
−11.86615952412879883675120220334, −11.81769245593984046772221818610, −11.07309711780990470135498774543, −10.90049568588655448578416361375, −10.44045813142318194879587015096, −9.876844696879021206733400526336, −9.398463637190622770551158001881, −8.615496669041581633535380365428, −8.028194353469524197375006004341, −8.017085093716133066922387052803, −7.22767067708817501952052441572, −6.55099079633660657159301651624, −5.98640784741726343744085534012, −5.64592705785796877397900285771, −4.96667887408346398049981481962, −4.66183592269911041341729648264, −3.65563113816889055198977811265, −3.25509334579925175795641258632, −1.52003660745108372056550483693, −1.29477599766883490247253829196,
1.29477599766883490247253829196, 1.52003660745108372056550483693, 3.25509334579925175795641258632, 3.65563113816889055198977811265, 4.66183592269911041341729648264, 4.96667887408346398049981481962, 5.64592705785796877397900285771, 5.98640784741726343744085534012, 6.55099079633660657159301651624, 7.22767067708817501952052441572, 8.017085093716133066922387052803, 8.028194353469524197375006004341, 8.615496669041581633535380365428, 9.398463637190622770551158001881, 9.876844696879021206733400526336, 10.44045813142318194879587015096, 10.90049568588655448578416361375, 11.07309711780990470135498774543, 11.81769245593984046772221818610, 11.86615952412879883675120220334
