L(s) = 1 | + 2·3-s + 9-s + 12·11-s + 4·13-s − 12·23-s − 6·25-s − 4·27-s + 24·33-s + 12·37-s + 8·39-s − 10·49-s + 16·59-s + 28·61-s − 24·69-s − 4·71-s − 28·73-s − 12·75-s − 11·81-s − 16·83-s + 4·97-s + 12·99-s + 36·107-s + 12·109-s + 24·111-s + 4·117-s + 86·121-s + 127-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1/3·9-s + 3.61·11-s + 1.10·13-s − 2.50·23-s − 6/5·25-s − 0.769·27-s + 4.17·33-s + 1.97·37-s + 1.28·39-s − 1.42·49-s + 2.08·59-s + 3.58·61-s − 2.88·69-s − 0.474·71-s − 3.27·73-s − 1.38·75-s − 1.22·81-s − 1.75·83-s + 0.406·97-s + 1.20·99-s + 3.48·107-s + 1.14·109-s + 2.27·111-s + 0.369·117-s + 7.81·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.482048495\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.482048495\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.878376583314565709515988150650, −8.331420707616764826583042598870, −8.119321761242708900614562817168, −7.35622902308931120005953550776, −6.95556909204739164167709631680, −6.32076764353867371001165550609, −5.94691251018865401690641595052, −5.84493337287348416539622628292, −4.42426362198009611408546968702, −4.14991391129811641343695020033, −3.62388232922820390627798777230, −3.59025133227290315875645783201, −2.37082491510612846557132007360, −1.78722102085145557690346531679, −1.14755912620769522242228089263,
1.14755912620769522242228089263, 1.78722102085145557690346531679, 2.37082491510612846557132007360, 3.59025133227290315875645783201, 3.62388232922820390627798777230, 4.14991391129811641343695020033, 4.42426362198009611408546968702, 5.84493337287348416539622628292, 5.94691251018865401690641595052, 6.32076764353867371001165550609, 6.95556909204739164167709631680, 7.35622902308931120005953550776, 8.119321761242708900614562817168, 8.331420707616764826583042598870, 8.878376583314565709515988150650