| L(s) = 1 | + 3·4-s − 2·5-s + 5·16-s − 6·20-s − 25-s + 2·29-s + 8·31-s − 20·41-s + 10·49-s + 8·59-s + 4·61-s + 3·64-s + 16·71-s − 16·79-s − 10·80-s + 20·89-s − 3·100-s − 4·101-s + 28·109-s + 6·116-s − 22·121-s + 24·124-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 0.894·5-s + 5/4·16-s − 1.34·20-s − 1/5·25-s + 0.371·29-s + 1.43·31-s − 3.12·41-s + 10/7·49-s + 1.04·59-s + 0.512·61-s + 3/8·64-s + 1.89·71-s − 1.80·79-s − 1.11·80-s + 2.11·89-s − 0.299·100-s − 0.398·101-s + 2.68·109-s + 0.557·116-s − 2·121-s + 2.15·124-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1703025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1703025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.536118724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.536118724\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 29 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−10.00505846949518651350295593141, −9.673208401878116675694158808760, −8.773355966290434546116607870882, −8.485499092341623796456950824390, −8.358543160738460753766710413005, −7.60242132995005891459753887582, −7.46510970093759013940925842897, −7.00451176912474970652342581349, −6.59130344346890757875140372092, −6.28641752581960476903018400151, −5.84487695249309432141090844679, −5.03942382321404709281914026419, −4.98732694313822228938769051418, −4.10809068241645342297028270241, −3.71728796090219176312261600033, −3.23711203771078452186518527739, −2.67440266786593693114220106013, −2.17323644387502277931074269001, −1.55265502428620737703863827011, −0.65504233145492505097997711043,
0.65504233145492505097997711043, 1.55265502428620737703863827011, 2.17323644387502277931074269001, 2.67440266786593693114220106013, 3.23711203771078452186518527739, 3.71728796090219176312261600033, 4.10809068241645342297028270241, 4.98732694313822228938769051418, 5.03942382321404709281914026419, 5.84487695249309432141090844679, 6.28641752581960476903018400151, 6.59130344346890757875140372092, 7.00451176912474970652342581349, 7.46510970093759013940925842897, 7.60242132995005891459753887582, 8.358543160738460753766710413005, 8.485499092341623796456950824390, 8.773355966290434546116607870882, 9.673208401878116675694158808760, 10.00505846949518651350295593141
