| L(s) = 1 | + 2·3-s − 2·4-s + 3·5-s + 9-s − 4·12-s − 13-s + 6·15-s + 4·16-s − 6·20-s + 4·25-s − 4·27-s − 8·31-s − 2·36-s + 14·37-s − 2·39-s + 6·41-s + 14·43-s + 3·45-s + 8·48-s + 5·49-s + 2·52-s + 18·53-s − 12·60-s − 8·64-s − 3·65-s − 10·67-s − 6·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4-s + 1.34·5-s + 1/3·9-s − 1.15·12-s − 0.277·13-s + 1.54·15-s + 16-s − 1.34·20-s + 4/5·25-s − 0.769·27-s − 1.43·31-s − 1/3·36-s + 2.30·37-s − 0.320·39-s + 0.937·41-s + 2.13·43-s + 0.447·45-s + 1.15·48-s + 5/7·49-s + 0.277·52-s + 2.47·53-s − 1.54·60-s − 64-s − 0.372·65-s − 1.22·67-s − 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.350598574\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.350598574\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 61 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−9.295455792544398235160794241855, −8.945278920230843320013949158476, −8.753475712992221271614078033529, −7.898917725872179517304083979808, −7.59970265412245632585978712072, −7.16053885305706772719092680227, −6.07144536313874732883591372241, −5.90088707643617630662279975277, −5.41223711946382281300473715397, −4.63429937210472043173403028075, −4.06298243951686871869876247721, −3.52719580513896865585186872763, −2.49374780566361831163582074504, −2.35694998327777122741685618488, −1.09218051133920291771885075771,
1.09218051133920291771885075771, 2.35694998327777122741685618488, 2.49374780566361831163582074504, 3.52719580513896865585186872763, 4.06298243951686871869876247721, 4.63429937210472043173403028075, 5.41223711946382281300473715397, 5.90088707643617630662279975277, 6.07144536313874732883591372241, 7.16053885305706772719092680227, 7.59970265412245632585978712072, 7.898917725872179517304083979808, 8.753475712992221271614078033529, 8.945278920230843320013949158476, 9.295455792544398235160794241855
