| L(s) = 1 | + 2·9-s + 8·11-s + 2·19-s − 4·29-s + 16·31-s + 12·41-s − 2·49-s + 4·61-s + 16·71-s + 8·79-s − 5·81-s − 28·89-s + 16·99-s + 20·101-s + 4·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 4·171-s + ⋯ |
| L(s) = 1 | + 2/3·9-s + 2.41·11-s + 0.458·19-s − 0.742·29-s + 2.87·31-s + 1.87·41-s − 2/7·49-s + 0.512·61-s + 1.89·71-s + 0.900·79-s − 5/9·81-s − 2.96·89-s + 1.60·99-s + 1.99·101-s + 0.383·109-s + 2.36·121-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 + 0.769·169-s + 0.305·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14440000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.402907649\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.402907649\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−9.062637888952616499398035585922, −8.393238122637518363411303984260, −7.992466461446165197499594741279, −7.39314109264119790576071992967, −7.30804555659862462551658221561, −6.79659774327782987305777387436, −6.36826400948377149712578542187, −6.27470248782849978982488381906, −5.87314707842815707053272046597, −5.27633065549417652667983947116, −4.78371975677658411754271284560, −4.43399402317373315832770699039, −4.11979998859652790974977929155, −3.65548339228154982902750704192, −3.43721839330477172859855161501, −2.57469427640427796331503080859, −2.38774208405392938514859790285, −1.45884125301415452683789816744, −1.22950003105022720702682018117, −0.70930286712597361521081827155,
0.70930286712597361521081827155, 1.22950003105022720702682018117, 1.45884125301415452683789816744, 2.38774208405392938514859790285, 2.57469427640427796331503080859, 3.43721839330477172859855161501, 3.65548339228154982902750704192, 4.11979998859652790974977929155, 4.43399402317373315832770699039, 4.78371975677658411754271284560, 5.27633065549417652667983947116, 5.87314707842815707053272046597, 6.27470248782849978982488381906, 6.36826400948377149712578542187, 6.79659774327782987305777387436, 7.30804555659862462551658221561, 7.39314109264119790576071992967, 7.992466461446165197499594741279, 8.393238122637518363411303984260, 9.062637888952616499398035585922
