| L(s) = 1 | + 2·3-s + 3·9-s + 4·13-s + 6·17-s + 6·23-s − 25-s + 4·27-s − 12·29-s + 8·39-s + 20·43-s + 5·49-s + 12·51-s − 6·53-s + 2·61-s + 12·69-s − 2·75-s − 2·79-s + 5·81-s − 24·87-s + 36·101-s + 8·103-s + 18·107-s + 36·113-s + 12·117-s + 13·121-s + 127-s + 40·129-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s + 1.10·13-s + 1.45·17-s + 1.25·23-s − 1/5·25-s + 0.769·27-s − 2.22·29-s + 1.28·39-s + 3.04·43-s + 5/7·49-s + 1.68·51-s − 0.824·53-s + 0.256·61-s + 1.44·69-s − 0.230·75-s − 0.225·79-s + 5/9·81-s − 2.57·87-s + 3.58·101-s + 0.788·103-s + 1.74·107-s + 3.38·113-s + 1.10·117-s + 1.18·121-s + 0.0887·127-s + 3.52·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.782401807\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.782401807\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 113 T^{2} + p^{2} T^{4} \) |
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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
−8.845026309621274272385248166211, −8.788143075197049700893508779004, −7.999768584385385841832577940530, −7.79651029857951105478143163827, −7.37536513455101635127577703700, −7.30760075688260827023128556939, −6.79166401962966952387495275289, −6.03343895032832643657582496434, −5.80019726504293163307167513017, −5.73514602796550376741728756764, −4.84635377285626084086237591097, −4.66039428943998520689860850489, −3.98205936988305686624600249845, −3.66198225429373975262115940561, −3.28454981012548143159963154571, −3.03966472605912994544838738812, −2.13952018937815109958602464180, −2.05297207700129830506206920802, −1.14283586227593639859133010849, −0.803362586218383331631908939853,
0.803362586218383331631908939853, 1.14283586227593639859133010849, 2.05297207700129830506206920802, 2.13952018937815109958602464180, 3.03966472605912994544838738812, 3.28454981012548143159963154571, 3.66198225429373975262115940561, 3.98205936988305686624600249845, 4.66039428943998520689860850489, 4.84635377285626084086237591097, 5.73514602796550376741728756764, 5.80019726504293163307167513017, 6.03343895032832643657582496434, 6.79166401962966952387495275289, 7.30760075688260827023128556939, 7.37536513455101635127577703700, 7.79651029857951105478143163827, 7.999768584385385841832577940530, 8.788143075197049700893508779004, 8.845026309621274272385248166211
