| L(s) = 1 | + 2·3-s − 2·7-s + 9-s + 8·17-s − 4·21-s + 25-s − 4·27-s + 12·37-s − 4·41-s − 4·43-s + 4·47-s − 3·49-s + 16·51-s + 4·59-s − 2·63-s − 4·67-s + 2·75-s − 11·81-s + 4·83-s + 20·89-s + 4·109-s + 24·111-s − 16·119-s + 18·121-s − 8·123-s + 127-s − 8·129-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.755·7-s + 1/3·9-s + 1.94·17-s − 0.872·21-s + 1/5·25-s − 0.769·27-s + 1.97·37-s − 0.624·41-s − 0.609·43-s + 0.583·47-s − 3/7·49-s + 2.24·51-s + 0.520·59-s − 0.251·63-s − 0.488·67-s + 0.230·75-s − 1.22·81-s + 0.439·83-s + 2.11·89-s + 0.383·109-s + 2.27·111-s − 1.46·119-s + 1.63·121-s − 0.721·123-s + 0.0887·127-s − 0.704·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.778955429\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.778955429\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
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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
−8.237962126860533345356635067777, −7.963975437616185599944719641281, −7.48379068246643639997913477253, −7.20089370459761643458128038600, −6.50488663228933057170466504530, −6.02356955373733895032518093977, −5.72613191296571179173360736411, −5.04698804423157064890237017864, −4.54839975139416088813344914043, −3.74515255529219068277463917072, −3.48558599056114559093878963843, −2.98472364055334565841218575712, −2.50183779951154311150608437907, −1.71842259821615172246840069733, −0.78611020095572034434368316706,
0.78611020095572034434368316706, 1.71842259821615172246840069733, 2.50183779951154311150608437907, 2.98472364055334565841218575712, 3.48558599056114559093878963843, 3.74515255529219068277463917072, 4.54839975139416088813344914043, 5.04698804423157064890237017864, 5.72613191296571179173360736411, 6.02356955373733895032518093977, 6.50488663228933057170466504530, 7.20089370459761643458128038600, 7.48379068246643639997913477253, 7.963975437616185599944719641281, 8.237962126860533345356635067777
