| L(s) = 1 | − 9-s − 4·11-s + 16·19-s − 2·29-s + 18·31-s + 14·41-s − 49-s − 30·59-s + 14·61-s − 24·71-s − 20·79-s + 81-s − 12·89-s + 4·99-s + 8·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s − 16·171-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 1.20·11-s + 3.67·19-s − 0.371·29-s + 3.23·31-s + 2.18·41-s − 1/7·49-s − 3.90·59-s + 1.79·61-s − 2.84·71-s − 2.25·79-s + 1/9·81-s − 1.27·89-s + 0.402·99-s + 0.766·109-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1/13·169-s − 1.22·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.531521618\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.531521618\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 63 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 165 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−8.913067288160459893498562448786, −8.046854527098847275014999972790, −7.77714267169910496054894008324, −7.55583067110813527162127505248, −7.36348948668968316729332993587, −6.80831945039338286636668479829, −6.32713222566845529569125232255, −5.87712503052612704630824548924, −5.63057711604380933582371233559, −5.36789277713461184616669776986, −4.83155589201491386930490059621, −4.45075956856058825818274409401, −4.25062579801762540719422130078, −3.27501198139940463956633060584, −3.13041896832936259124440653491, −2.78127621000326369164773190579, −2.50580839908929769018362274729, −1.46080762879925085645680287446, −1.17340843254194289101868941446, −0.50349130634043415675373723900,
0.50349130634043415675373723900, 1.17340843254194289101868941446, 1.46080762879925085645680287446, 2.50580839908929769018362274729, 2.78127621000326369164773190579, 3.13041896832936259124440653491, 3.27501198139940463956633060584, 4.25062579801762540719422130078, 4.45075956856058825818274409401, 4.83155589201491386930490059621, 5.36789277713461184616669776986, 5.63057711604380933582371233559, 5.87712503052612704630824548924, 6.32713222566845529569125232255, 6.80831945039338286636668479829, 7.36348948668968316729332993587, 7.55583067110813527162127505248, 7.77714267169910496054894008324, 8.046854527098847275014999972790, 8.913067288160459893498562448786
