| L(s) = 1 | − 3·9-s + 4·11-s − 6·13-s − 4·16-s − 5·17-s + 6·19-s − 4·23-s + 25-s + 12·37-s − 12·41-s − 7·49-s − 4·53-s − 12·61-s + 10·67-s − 8·71-s − 6·73-s − 12·83-s − 12·99-s − 6·101-s − 20·113-s + 18·117-s + 5·121-s + 127-s + 131-s + 137-s + 139-s − 24·143-s + ⋯ |
| L(s) = 1 | − 9-s + 1.20·11-s − 1.66·13-s − 16-s − 1.21·17-s + 1.37·19-s − 0.834·23-s + 1/5·25-s + 1.97·37-s − 1.87·41-s − 49-s − 0.549·53-s − 1.53·61-s + 1.22·67-s − 0.949·71-s − 0.702·73-s − 1.31·83-s − 1.20·99-s − 0.597·101-s − 1.88·113-s + 1.66·117-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.00·143-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100793 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100793 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | $C_2$ | \( 1 + p T^{2} \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 6 T + p T^{2} ) \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 65 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
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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
−9.338454283531721201393266967225, −8.974044271352804509584049573607, −8.323251111737686294752321749541, −7.909876394959228313322877501788, −7.19158236917613228360900327390, −6.86523467170533718035548845884, −6.29726549353495472853486588355, −5.78370835039020955277033303146, −5.00896666513353621837480702638, −4.61543466662304898150771612711, −4.01944433862736986018637679779, −3.06440983181705312700028765882, −2.58019999941638056307726197773, −1.67093919381020811892559199023, 0,
1.67093919381020811892559199023, 2.58019999941638056307726197773, 3.06440983181705312700028765882, 4.01944433862736986018637679779, 4.61543466662304898150771612711, 5.00896666513353621837480702638, 5.78370835039020955277033303146, 6.29726549353495472853486588355, 6.86523467170533718035548845884, 7.19158236917613228360900327390, 7.909876394959228313322877501788, 8.323251111737686294752321749541, 8.974044271352804509584049573607, 9.338454283531721201393266967225
