| L(s) = 1 | + 2-s − 2·7-s − 8-s + 3·9-s + 10·11-s + 2·13-s − 2·14-s − 16-s + 3·18-s − 19-s + 10·22-s + 23-s + 2·26-s − 6·29-s + 8·31-s + 22·37-s − 38-s + 9·41-s + 6·43-s + 46-s − 11·49-s + 5·53-s + 2·56-s − 6·58-s + 8·62-s − 6·63-s + 64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.755·7-s − 0.353·8-s + 9-s + 3.01·11-s + 0.554·13-s − 0.534·14-s − 1/4·16-s + 0.707·18-s − 0.229·19-s + 2.13·22-s + 0.208·23-s + 0.392·26-s − 1.11·29-s + 1.43·31-s + 3.61·37-s − 0.162·38-s + 1.40·41-s + 0.914·43-s + 0.147·46-s − 1.57·49-s + 0.686·53-s + 0.267·56-s − 0.787·58-s + 1.01·62-s − 0.755·63-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.975409050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.975409050\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 6 T - 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 5 T - 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 7 T - 40 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−9.932918427995837414089519391319, −9.719832933752611909378027284309, −9.430729454746527553327742374237, −9.317519993778239477508888710175, −8.447918354250738830884132284406, −8.433991680114459480987162768451, −7.40655059846664527146659403522, −7.35155301191210300128862824800, −6.51779602412075469500115003179, −6.41831436480062518115554336333, −6.11675240902167177274167002985, −5.64848032313846357543304691273, −4.73339480470200908725662950753, −4.29353267796000027187381777841, −3.92842817128213055915857811861, −3.87614472238841220322712760618, −2.98054166850019483021196192632, −2.40931606491552331823877825361, −1.29793947836325581910441911715, −1.04987745002552233066082512272,
1.04987745002552233066082512272, 1.29793947836325581910441911715, 2.40931606491552331823877825361, 2.98054166850019483021196192632, 3.87614472238841220322712760618, 3.92842817128213055915857811861, 4.29353267796000027187381777841, 4.73339480470200908725662950753, 5.64848032313846357543304691273, 6.11675240902167177274167002985, 6.41831436480062518115554336333, 6.51779602412075469500115003179, 7.35155301191210300128862824800, 7.40655059846664527146659403522, 8.433991680114459480987162768451, 8.447918354250738830884132284406, 9.317519993778239477508888710175, 9.430729454746527553327742374237, 9.719832933752611909378027284309, 9.932918427995837414089519391319
