| L(s) = 1 | − 2·3-s + 2·5-s + 7-s + 9-s − 4·15-s − 2·17-s − 2·21-s − 6·25-s + 4·27-s + 2·35-s − 8·37-s − 10·41-s + 4·43-s + 2·45-s + 8·47-s + 49-s + 4·51-s − 4·59-s + 63-s − 4·67-s + 12·75-s + 4·79-s − 11·81-s + 20·83-s − 4·85-s + 22·89-s + 2·101-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.03·15-s − 0.485·17-s − 0.436·21-s − 6/5·25-s + 0.769·27-s + 0.338·35-s − 1.31·37-s − 1.56·41-s + 0.609·43-s + 0.298·45-s + 1.16·47-s + 1/7·49-s + 0.560·51-s − 0.520·59-s + 0.125·63-s − 0.488·67-s + 1.38·75-s + 0.450·79-s − 1.22·81-s + 2.19·83-s − 0.433·85-s + 2.33·89-s + 0.199·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
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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
−7.947125111037089851561052451027, −7.54625508361084281186226476498, −7.03388022575249574307263875933, −6.54763624288520170395500080163, −6.06620641308968811689228022206, −5.92921189534370397921023669910, −5.22231978645475376411869284737, −5.04195840592919556518396932524, −4.50389273567438918384014055757, −3.78253384244254093730748973159, −3.29867775593355626285410384967, −2.29633111132606443586769264931, −1.98835518282801419264155036880, −1.10694203396887396541640129759, 0,
1.10694203396887396541640129759, 1.98835518282801419264155036880, 2.29633111132606443586769264931, 3.29867775593355626285410384967, 3.78253384244254093730748973159, 4.50389273567438918384014055757, 5.04195840592919556518396932524, 5.22231978645475376411869284737, 5.92921189534370397921023669910, 6.06620641308968811689228022206, 6.54763624288520170395500080163, 7.03388022575249574307263875933, 7.54625508361084281186226476498, 7.947125111037089851561052451027
