L(s) = 1 | + 2·3-s + 9-s − 8·11-s − 4·17-s + 10·19-s − 2·25-s − 4·27-s − 16·33-s + 12·41-s + 12·43-s − 49-s − 8·51-s + 20·57-s − 14·59-s + 4·67-s − 12·73-s − 4·75-s − 11·81-s + 10·83-s + 8·89-s + 16·97-s − 8·99-s + 8·107-s + 4·113-s + 26·121-s + 24·123-s + 127-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1/3·9-s − 2.41·11-s − 0.970·17-s + 2.29·19-s − 2/5·25-s − 0.769·27-s − 2.78·33-s + 1.87·41-s + 1.82·43-s − 1/7·49-s − 1.12·51-s + 2.64·57-s − 1.82·59-s + 0.488·67-s − 1.40·73-s − 0.461·75-s − 1.22·81-s + 1.09·83-s + 0.847·89-s + 1.62·97-s − 0.804·99-s + 0.773·107-s + 0.376·113-s + 2.36·121-s + 2.16·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.220980163\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.220980163\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
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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
−7.911384428652507507043545201130, −7.71707648216405982289174654163, −7.62848982239353070223355267844, −7.18867557136600453153120983152, −6.36095350767963730598708193030, −5.76914760177565068515307738376, −5.57267820004726959504034413630, −4.92858663461306474391830432005, −4.55913975844499161484477067696, −3.85558483155737932819455366337, −3.24462995980649300292507639481, −2.74536329980872402684645925371, −2.53704876383508402875927515466, −1.80898076070864021850494901114, −0.63069742471219787285167065443,
0.63069742471219787285167065443, 1.80898076070864021850494901114, 2.53704876383508402875927515466, 2.74536329980872402684645925371, 3.24462995980649300292507639481, 3.85558483155737932819455366337, 4.55913975844499161484477067696, 4.92858663461306474391830432005, 5.57267820004726959504034413630, 5.76914760177565068515307738376, 6.36095350767963730598708193030, 7.18867557136600453153120983152, 7.62848982239353070223355267844, 7.71707648216405982289174654163, 7.911384428652507507043545201130