| L(s) = 1 | + 3-s + 5-s + 7-s + 3·9-s + 2·11-s + 8·13-s + 15-s − 6·19-s + 21-s − 3·23-s + 8·27-s − 6·29-s + 2·33-s + 35-s + 12·37-s + 8·39-s − 14·41-s − 18·43-s + 3·45-s − 6·49-s + 6·53-s + 2·55-s − 6·57-s + 10·59-s − 5·61-s + 3·63-s + 8·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 9-s + 0.603·11-s + 2.21·13-s + 0.258·15-s − 1.37·19-s + 0.218·21-s − 0.625·23-s + 1.53·27-s − 1.11·29-s + 0.348·33-s + 0.169·35-s + 1.97·37-s + 1.28·39-s − 2.18·41-s − 2.74·43-s + 0.447·45-s − 6/7·49-s + 0.824·53-s + 0.269·55-s − 0.794·57-s + 1.30·59-s − 0.640·61-s + 0.377·63-s + 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.342797800\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.342797800\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 8 T - 9 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 6 T - 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 17 T + 200 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−11.84209759969877538035208474160, −11.75937320688817184309430848931, −11.08944934396701605930796412373, −10.68744271651203120512119314324, −10.01621194868901967206026104277, −9.969614304350331885090936122221, −9.085795471971761816262852933449, −8.723048765687327553149368847023, −8.254349299597378995482278508418, −8.089326253090304074177819828932, −6.93908139014821479229079562924, −6.85581568554923371841397739720, −6.06103251015962913007064911157, −5.81835868962309026392036215833, −4.75173078951368002675729806368, −4.28594130121994344478204656725, −3.69355667433550680345315236611, −3.08037391997683798988042195968, −1.83232355697355770878513762232, −1.48216594520411328352741169624,
1.48216594520411328352741169624, 1.83232355697355770878513762232, 3.08037391997683798988042195968, 3.69355667433550680345315236611, 4.28594130121994344478204656725, 4.75173078951368002675729806368, 5.81835868962309026392036215833, 6.06103251015962913007064911157, 6.85581568554923371841397739720, 6.93908139014821479229079562924, 8.089326253090304074177819828932, 8.254349299597378995482278508418, 8.723048765687327553149368847023, 9.085795471971761816262852933449, 9.969614304350331885090936122221, 10.01621194868901967206026104277, 10.68744271651203120512119314324, 11.08944934396701605930796412373, 11.75937320688817184309430848931, 11.84209759969877538035208474160
