| L(s) = 1 | − 2·2-s + 3·4-s + 4·5-s − 5·7-s − 4·8-s − 8·10-s + 2·11-s − 6·13-s + 10·14-s + 5·16-s + 2·17-s + 4·19-s + 12·20-s − 4·22-s − 23-s + 5·25-s + 12·26-s − 15·28-s + 4·29-s − 18·31-s − 6·32-s − 4·34-s − 20·35-s − 8·37-s − 8·38-s − 16·40-s + 3·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 1.78·5-s − 1.88·7-s − 1.41·8-s − 2.52·10-s + 0.603·11-s − 1.66·13-s + 2.67·14-s + 5/4·16-s + 0.485·17-s + 0.917·19-s + 2.68·20-s − 0.852·22-s − 0.208·23-s + 25-s + 2.35·26-s − 2.83·28-s + 0.742·29-s − 3.23·31-s − 1.06·32-s − 0.685·34-s − 3.38·35-s − 1.31·37-s − 1.29·38-s − 2.52·40-s + 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6636507068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6636507068\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 14 T + 113 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 10 T + 3 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−10.09832193012939156029554991694, −9.512071200893393874310744404821, −9.352579939097450797621193429109, −8.919514443411919726073946605765, −8.919131551083318279840014129883, −7.84717689841616808754033586633, −7.42512854841746209680356384795, −7.22700567641195771352218044920, −6.78069913421670163184563675894, −6.36669513582243966388359006582, −5.86842606570837005200177743516, −5.48843525812126798449580445608, −5.36273239143794969526082966632, −4.27026930899992951753523723270, −3.62562183683325818788594680186, −2.91366157400800324719189310408, −2.77732063273569094578870802149, −1.88557386160367316324485437620, −1.61799698151306428413647275313, −0.42921544936964084669831979732,
0.42921544936964084669831979732, 1.61799698151306428413647275313, 1.88557386160367316324485437620, 2.77732063273569094578870802149, 2.91366157400800324719189310408, 3.62562183683325818788594680186, 4.27026930899992951753523723270, 5.36273239143794969526082966632, 5.48843525812126798449580445608, 5.86842606570837005200177743516, 6.36669513582243966388359006582, 6.78069913421670163184563675894, 7.22700567641195771352218044920, 7.42512854841746209680356384795, 7.84717689841616808754033586633, 8.919131551083318279840014129883, 8.919514443411919726073946605765, 9.352579939097450797621193429109, 9.512071200893393874310744404821, 10.09832193012939156029554991694
