| L(s) = 1 | − 2-s + 8-s − 3·9-s − 6·11-s + 2·13-s − 16-s − 3·17-s + 3·18-s − 19-s + 6·22-s − 12·23-s − 10·25-s − 2·26-s − 6·29-s − 4·31-s + 3·34-s + 4·37-s + 38-s + 9·41-s + 43-s + 12·46-s − 6·47-s + 10·50-s − 12·53-s + 6·58-s + 3·59-s + 8·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.353·8-s − 9-s − 1.80·11-s + 0.554·13-s − 1/4·16-s − 0.727·17-s + 0.707·18-s − 0.229·19-s + 1.27·22-s − 2.50·23-s − 2·25-s − 0.392·26-s − 1.11·29-s − 0.718·31-s + 0.514·34-s + 0.657·37-s + 0.162·38-s + 1.40·41-s + 0.152·43-s + 1.76·46-s − 0.875·47-s + 1.41·50-s − 1.64·53-s + 0.787·58-s + 0.390·59-s + 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.826482394915270983877045632761, −9.512870047397212830119143199975, −9.170484032867806589778505471899, −8.610605577023071918986177851062, −8.102058923452535161213033571947, −7.88502748715038301407415213909, −7.74042725857278720409434091582, −7.15535676633258955492244607527, −6.14867184707126846109107272073, −6.12571067506822676859314795739, −5.71703849630535701426693235105, −5.12957395228249303880997085880, −4.59558300352595170361857930128, −3.76948062762121412111966361272, −3.73594527000375047896000661127, −2.58014116395012826664060865930, −2.31124307550230393269595319017, −1.66100292459587211726408205633, 0, 0,
1.66100292459587211726408205633, 2.31124307550230393269595319017, 2.58014116395012826664060865930, 3.73594527000375047896000661127, 3.76948062762121412111966361272, 4.59558300352595170361857930128, 5.12957395228249303880997085880, 5.71703849630535701426693235105, 6.12571067506822676859314795739, 6.14867184707126846109107272073, 7.15535676633258955492244607527, 7.74042725857278720409434091582, 7.88502748715038301407415213909, 8.102058923452535161213033571947, 8.610605577023071918986177851062, 9.170484032867806589778505471899, 9.512870047397212830119143199975, 9.826482394915270983877045632761
