| L(s) = 1 | − 2-s − 3·5-s + 5·7-s + 8-s + 3·10-s + 6·11-s + 2·13-s − 5·14-s − 16-s + 6·17-s − 2·19-s − 6·22-s + 5·25-s − 2·26-s − 3·29-s − 5·31-s − 6·34-s − 15·35-s + 10·37-s + 2·38-s − 3·40-s − 9·41-s − 5·43-s + 3·47-s + 18·49-s − 5·50-s + 9·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.34·5-s + 1.88·7-s + 0.353·8-s + 0.948·10-s + 1.80·11-s + 0.554·13-s − 1.33·14-s − 1/4·16-s + 1.45·17-s − 0.458·19-s − 1.27·22-s + 25-s − 0.392·26-s − 0.557·29-s − 0.898·31-s − 1.02·34-s − 2.53·35-s + 1.64·37-s + 0.324·38-s − 0.474·40-s − 1.40·41-s − 0.762·43-s + 0.437·47-s + 18/7·49-s − 0.707·50-s + 1.23·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.022943142\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.022943142\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 15 T + 154 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
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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
−9.558524482135698843826502140437, −8.940296143253736087997133739084, −8.660161343308378109829011707465, −8.461126292088583886472329142911, −7.991544528823111430466237661566, −7.85014358737131025691059955794, −7.14742900066598857195594652205, −7.12943241991317322691095699385, −6.54783309528024626315345954962, −5.89200493959042981610443686874, −5.39314858132892201430648652865, −5.11020833972679938390114143946, −4.39952692151742650843801234170, −4.12687235452333438506841260754, −3.68597568018065319908927841122, −3.46508541704207573952184220096, −2.36065781263820098087124220641, −1.79061759356413652485099787565, −1.14572372312448390682118497022, −0.77844600995590594592811077195,
0.77844600995590594592811077195, 1.14572372312448390682118497022, 1.79061759356413652485099787565, 2.36065781263820098087124220641, 3.46508541704207573952184220096, 3.68597568018065319908927841122, 4.12687235452333438506841260754, 4.39952692151742650843801234170, 5.11020833972679938390114143946, 5.39314858132892201430648652865, 5.89200493959042981610443686874, 6.54783309528024626315345954962, 7.12943241991317322691095699385, 7.14742900066598857195594652205, 7.85014358737131025691059955794, 7.991544528823111430466237661566, 8.461126292088583886472329142911, 8.660161343308378109829011707465, 8.940296143253736087997133739084, 9.558524482135698843826502140437
