L(s) = 1 | + 3·5-s + 8·11-s + 5·13-s − 5·17-s + 8·19-s − 23-s + 5·25-s + 3·29-s + 8·31-s + 4·37-s − 5·41-s + 11·43-s − 5·47-s − 14·49-s − 9·53-s + 24·55-s + 13·59-s + 61-s + 15·65-s + 5·67-s + 71-s + 9·73-s − 17·79-s − 32·83-s − 15·85-s + 3·89-s + 24·95-s + ⋯ |
L(s) = 1 | + 1.34·5-s + 2.41·11-s + 1.38·13-s − 1.21·17-s + 1.83·19-s − 0.208·23-s + 25-s + 0.557·29-s + 1.43·31-s + 0.657·37-s − 0.780·41-s + 1.67·43-s − 0.729·47-s − 2·49-s − 1.23·53-s + 3.23·55-s + 1.69·59-s + 0.128·61-s + 1.86·65-s + 0.610·67-s + 0.118·71-s + 1.05·73-s − 1.91·79-s − 3.51·83-s − 1.62·85-s + 0.317·89-s + 2.46·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.518147496\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.518147496\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 5 T + 8 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 5 T - 22 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 13 T + 110 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - T - 70 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 9 T + 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
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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
−9.749983469200895253463882489851, −9.367681820947984554783722188979, −8.991779058125508390947565010689, −8.828028912624420470127963299747, −8.248787913468227015034173976083, −7.928806002141256890635426187922, −7.11649782786393718343444923454, −6.80868988005879988007847041521, −6.34505723505851555093738568329, −6.29180913169881659201104257972, −5.77633603010819234277034443891, −5.30427868616545111350931758606, −4.51418192937962013112816625171, −4.44977473391486715702264048887, −3.64121172677040595013894719947, −3.32855668736504267473873279740, −2.65401172272635213634698606324, −1.95471061394571443070945452840, −1.19434886791347851552481812535, −1.16206996123835915376356632410,
1.16206996123835915376356632410, 1.19434886791347851552481812535, 1.95471061394571443070945452840, 2.65401172272635213634698606324, 3.32855668736504267473873279740, 3.64121172677040595013894719947, 4.44977473391486715702264048887, 4.51418192937962013112816625171, 5.30427868616545111350931758606, 5.77633603010819234277034443891, 6.29180913169881659201104257972, 6.34505723505851555093738568329, 6.80868988005879988007847041521, 7.11649782786393718343444923454, 7.928806002141256890635426187922, 8.248787913468227015034173976083, 8.828028912624420470127963299747, 8.991779058125508390947565010689, 9.367681820947984554783722188979, 9.749983469200895253463882489851