| L(s) = 1 | − 2·2-s + 4-s + 4·7-s + 4·11-s + 4·13-s − 8·14-s + 16-s − 4·19-s − 8·22-s − 12·23-s − 8·26-s + 4·28-s − 2·29-s − 4·31-s + 2·32-s + 8·38-s + 12·41-s + 12·43-s + 4·44-s + 24·46-s − 12·47-s + 6·49-s + 4·52-s + 4·53-s + 4·58-s + 4·61-s + 8·62-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s + 1.51·7-s + 1.20·11-s + 1.10·13-s − 2.13·14-s + 1/4·16-s − 0.917·19-s − 1.70·22-s − 2.50·23-s − 1.56·26-s + 0.755·28-s − 0.371·29-s − 0.718·31-s + 0.353·32-s + 1.29·38-s + 1.87·41-s + 1.82·43-s + 0.603·44-s + 3.53·46-s − 1.75·47-s + 6/7·49-s + 0.554·52-s + 0.549·53-s + 0.525·58-s + 0.512·61-s + 1.01·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42575625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42575625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.649900238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.649900238\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_4$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 34 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 + 12 T + 74 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + 4 T - 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 94 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 20 T + 258 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 138 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.145519897957742037803230067097, −7.989564771931177069470950026661, −7.71553176950560475814563697229, −7.46325394231910153867160895862, −6.73170715204644303945319137399, −6.35163232845224554527227418260, −6.31686255870775567119728404287, −5.78638216961660261835052815271, −5.46384112041277478470517063406, −4.98324011054116642298086614234, −4.45819293555289265652110692436, −4.08398326576206621613594375929, −3.91777259507565956384099566673, −3.59330364098120732561622493720, −2.77897074812734334678037981653, −2.22382438145767870959389157923, −1.74682145759572670183904958864, −1.65682607629540478520366799219, −0.801284393212260615437203964955, −0.57104723809250590467170896966,
0.57104723809250590467170896966, 0.801284393212260615437203964955, 1.65682607629540478520366799219, 1.74682145759572670183904958864, 2.22382438145767870959389157923, 2.77897074812734334678037981653, 3.59330364098120732561622493720, 3.91777259507565956384099566673, 4.08398326576206621613594375929, 4.45819293555289265652110692436, 4.98324011054116642298086614234, 5.46384112041277478470517063406, 5.78638216961660261835052815271, 6.31686255870775567119728404287, 6.35163232845224554527227418260, 6.73170715204644303945319137399, 7.46325394231910153867160895862, 7.71553176950560475814563697229, 7.989564771931177069470950026661, 8.145519897957742037803230067097
