| L(s) = 1 | + 2·9-s + 10·11-s − 2·19-s − 18·29-s − 4·31-s + 6·41-s + 5·49-s + 12·59-s − 20·61-s + 4·71-s + 22·79-s − 5·81-s + 28·89-s + 20·99-s − 28·101-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 + 169-s + ⋯ |
| L(s) = 1 | + 2/3·9-s + 3.01·11-s − 0.458·19-s − 3.34·29-s − 0.718·31-s + 0.937·41-s + 5/7·49-s + 1.56·59-s − 2.56·61-s + 0.474·71-s + 2.47·79-s − 5/9·81-s + 2.96·89-s + 2.01·99-s − 2.78·101-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 + 1/13·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.163909928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.163909928\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 23 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−8.909066322925044918388895541115, −8.080804763659231577310421259525, −7.63822396663609453925233187010, −7.27927731261662157156767611314, −7.22330535884535179799266917387, −6.67033661628611123229130692787, −6.24038310374922963070809859646, −6.09450216737845948746616268209, −5.76136297393160432215938394070, −4.99545581530395674895248766149, −4.91478241195045360768129282662, −4.08000153979078475101532858048, −3.99802892059942315797710506709, −3.60907052529555547302216594091, −3.53633131033904080334257543496, −2.49694239142338475435054588618, −2.09013074114321655717971329401, −1.51594355725472460931701144108, −1.33932896517752793247981020920, −0.50674597202377969495523933350,
0.50674597202377969495523933350, 1.33932896517752793247981020920, 1.51594355725472460931701144108, 2.09013074114321655717971329401, 2.49694239142338475435054588618, 3.53633131033904080334257543496, 3.60907052529555547302216594091, 3.99802892059942315797710506709, 4.08000153979078475101532858048, 4.91478241195045360768129282662, 4.99545581530395674895248766149, 5.76136297393160432215938394070, 6.09450216737845948746616268209, 6.24038310374922963070809859646, 6.67033661628611123229130692787, 7.22330535884535179799266917387, 7.27927731261662157156767611314, 7.63822396663609453925233187010, 8.080804763659231577310421259525, 8.909066322925044918388895541115
