| L(s) = 1 | + 1.80·2-s + 3-s + 1.25·4-s + 2.77·5-s + 1.80·6-s − 7-s − 1.33·8-s + 9-s + 5.01·10-s + 1.25·12-s + 2.31·13-s − 1.80·14-s + 2.77·15-s − 4.93·16-s + 2.66·17-s + 1.80·18-s + 8.08·19-s + 3.49·20-s − 21-s − 2.43·23-s − 1.33·24-s + 2.71·25-s + 4.17·26-s + 27-s − 1.25·28-s − 7.55·29-s + 5.01·30-s + ⋯ |
| L(s) = 1 | + 1.27·2-s + 0.577·3-s + 0.629·4-s + 1.24·5-s + 0.737·6-s − 0.377·7-s − 0.472·8-s + 0.333·9-s + 1.58·10-s + 0.363·12-s + 0.641·13-s − 0.482·14-s + 0.717·15-s − 1.23·16-s + 0.645·17-s + 0.425·18-s + 1.85·19-s + 0.782·20-s − 0.218·21-s − 0.506·23-s − 0.272·24-s + 0.542·25-s + 0.819·26-s + 0.192·27-s − 0.238·28-s − 1.40·29-s + 0.915·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.235830280\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.235830280\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.80T + 2T^{2} \) |
| 5 | \( 1 - 2.77T + 5T^{2} \) |
| 13 | \( 1 - 2.31T + 13T^{2} \) |
| 17 | \( 1 - 2.66T + 17T^{2} \) |
| 19 | \( 1 - 8.08T + 19T^{2} \) |
| 23 | \( 1 + 2.43T + 23T^{2} \) |
| 29 | \( 1 + 7.55T + 29T^{2} \) |
| 31 | \( 1 - 9.24T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 + 0.299T + 41T^{2} \) |
| 43 | \( 1 + 7.29T + 43T^{2} \) |
| 47 | \( 1 - 0.457T + 47T^{2} \) |
| 53 | \( 1 + 5.19T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 + 1.81T + 61T^{2} \) |
| 67 | \( 1 + 1.42T + 67T^{2} \) |
| 71 | \( 1 - 0.558T + 71T^{2} \) |
| 73 | \( 1 + 9.66T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 8.51T + 83T^{2} \) |
| 89 | \( 1 + 1.56T + 89T^{2} \) |
| 97 | \( 1 + 0.533T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.062065173598244791031719800744, −8.108399822942325593255229049329, −7.17831769397536614565056809878, −6.19443266069340295080261354778, −5.78181943719204635148760519313, −5.04219715019381701328102354021, −4.01982932837704398482115132361, −3.21050107610072853821125995608, −2.54772793470308109222937591042, −1.32630170646631392131259301264,
1.32630170646631392131259301264, 2.54772793470308109222937591042, 3.21050107610072853821125995608, 4.01982932837704398482115132361, 5.04219715019381701328102354021, 5.78181943719204635148760519313, 6.19443266069340295080261354778, 7.17831769397536614565056809878, 8.108399822942325593255229049329, 9.062065173598244791031719800744
