| L(s) = 1 | + 5·9-s − 10·11-s + 10·19-s + 8·29-s + 20·31-s + 10·41-s + 10·49-s + 20·61-s − 20·79-s + 16·81-s + 18·89-s − 50·99-s − 4·101-s + 20·109-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 50·171-s + 173-s + ⋯ |
| L(s) = 1 | + 5/3·9-s − 3.01·11-s + 2.29·19-s + 1.48·29-s + 3.59·31-s + 1.56·41-s + 10/7·49-s + 2.56·61-s − 2.25·79-s + 16/9·81-s + 1.90·89-s − 5.02·99-s − 0.398·101-s + 1.91·109-s + 4.81·121-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 + 2·169-s + 3.82·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.929433293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.929433293\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 125 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 121 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 165 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−9.885719314496483631087440279456, −9.444920249324078363851924348847, −8.489964967267012127252391952130, −8.451591241654617797928958535046, −8.011790149183488762133875175113, −7.48827396405676716769625049755, −7.31292880784136462291527510761, −7.08781247594143050208155581985, −6.27821204934432876523972304711, −5.99446293720170483428410644915, −5.34149140931135853662442739545, −5.09053976100896970407975199078, −4.54194332334893634143718979972, −4.47884914177644198254298165904, −3.57852601168250932625819451055, −3.00848023143031987805206135530, −2.53837244369318263664053504632, −2.33720387336688582995098144612, −0.988459762065002097456050266730, −0.899040366512349458854508529172,
0.899040366512349458854508529172, 0.988459762065002097456050266730, 2.33720387336688582995098144612, 2.53837244369318263664053504632, 3.00848023143031987805206135530, 3.57852601168250932625819451055, 4.47884914177644198254298165904, 4.54194332334893634143718979972, 5.09053976100896970407975199078, 5.34149140931135853662442739545, 5.99446293720170483428410644915, 6.27821204934432876523972304711, 7.08781247594143050208155581985, 7.31292880784136462291527510761, 7.48827396405676716769625049755, 8.011790149183488762133875175113, 8.451591241654617797928958535046, 8.489964967267012127252391952130, 9.444920249324078363851924348847, 9.885719314496483631087440279456
