| L(s) = 1 | + 6·9-s − 10·11-s + 12·19-s + 6·29-s + 4·31-s − 8·41-s − 49-s + 28·59-s + 8·61-s − 26·71-s − 2·79-s + 27·81-s − 20·89-s − 60·99-s + 24·101-s − 18·109-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + ⋯ |
| L(s) = 1 | + 2·9-s − 3.01·11-s + 2.75·19-s + 1.11·29-s + 0.718·31-s − 1.24·41-s − 1/7·49-s + 3.64·59-s + 1.02·61-s − 3.08·71-s − 0.225·79-s + 3·81-s − 2.11·89-s − 6.03·99-s + 2.38·101-s − 1.72·109-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 − 0.769·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.994934565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.994934565\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 125 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−10.39630201846709653688252958820, −10.10741351579027436891973055389, −9.898435165927122424821339861481, −9.833678509456435057677641150799, −8.880575765166552278343172025796, −8.396994660749056155037365720496, −7.986143326869911592311018799088, −7.56363556880996454325480902808, −7.11519169640077653996075052143, −7.10228689012684436915297422111, −6.22512540844802711355733326436, −5.49812720141374618627204639874, −5.06162779077396476297767527010, −5.05039575756716157407313005772, −4.26109177807554697867478453567, −3.61816846242806993735719031260, −2.86740551482613276423128265527, −2.60098483233759492199007751435, −1.60049408733384802633414554803, −0.790133013805134540606127223683,
0.790133013805134540606127223683, 1.60049408733384802633414554803, 2.60098483233759492199007751435, 2.86740551482613276423128265527, 3.61816846242806993735719031260, 4.26109177807554697867478453567, 5.05039575756716157407313005772, 5.06162779077396476297767527010, 5.49812720141374618627204639874, 6.22512540844802711355733326436, 7.10228689012684436915297422111, 7.11519169640077653996075052143, 7.56363556880996454325480902808, 7.986143326869911592311018799088, 8.396994660749056155037365720496, 8.880575765166552278343172025796, 9.833678509456435057677641150799, 9.898435165927122424821339861481, 10.10741351579027436891973055389, 10.39630201846709653688252958820
