| L(s) = 1 | + 0.0495·2-s − 3-s − 1.99·4-s − 5-s − 0.0495·6-s − 1.50·7-s − 0.197·8-s + 9-s − 0.0495·10-s + 1.36·11-s + 1.99·12-s − 6.59·13-s − 0.0746·14-s + 15-s + 3.98·16-s + 2.05·17-s + 0.0495·18-s + 4.23·19-s + 1.99·20-s + 1.50·21-s + 0.0676·22-s + 1.56·23-s + 0.197·24-s + 25-s − 0.326·26-s − 27-s + 3.01·28-s + ⋯ |
| L(s) = 1 | + 0.0350·2-s − 0.577·3-s − 0.998·4-s − 0.447·5-s − 0.0202·6-s − 0.569·7-s − 0.0699·8-s + 0.333·9-s − 0.0156·10-s + 0.411·11-s + 0.576·12-s − 1.82·13-s − 0.0199·14-s + 0.258·15-s + 0.996·16-s + 0.498·17-s + 0.0116·18-s + 0.971·19-s + 0.446·20-s + 0.328·21-s + 0.0144·22-s + 0.326·23-s + 0.0404·24-s + 0.200·25-s − 0.0640·26-s − 0.192·27-s + 0.569·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4251167591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4251167591\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 0.0495T + 2T^{2} \) |
| 7 | \( 1 + 1.50T + 7T^{2} \) |
| 11 | \( 1 - 1.36T + 11T^{2} \) |
| 13 | \( 1 + 6.59T + 13T^{2} \) |
| 17 | \( 1 - 2.05T + 17T^{2} \) |
| 19 | \( 1 - 4.23T + 19T^{2} \) |
| 23 | \( 1 - 1.56T + 23T^{2} \) |
| 29 | \( 1 + 6.29T + 29T^{2} \) |
| 31 | \( 1 + 2.44T + 31T^{2} \) |
| 37 | \( 1 + 6.90T + 37T^{2} \) |
| 41 | \( 1 + 8.13T + 41T^{2} \) |
| 43 | \( 1 - 4.49T + 43T^{2} \) |
| 47 | \( 1 - 1.58T + 47T^{2} \) |
| 53 | \( 1 + 14.0T + 53T^{2} \) |
| 59 | \( 1 + 5.18T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 - 6.50T + 67T^{2} \) |
| 71 | \( 1 + 4.56T + 71T^{2} \) |
| 73 | \( 1 + 4.68T + 73T^{2} \) |
| 79 | \( 1 - 0.561T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 + 5.25T + 89T^{2} \) |
| 97 | \( 1 - 9.91T + 97T^{2} \) |
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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
−7.83798066208240394023804992897, −7.48494528308985693665506200301, −6.71479890998734701163642680773, −5.78013002454355727943672360058, −5.06428545934688902540640276821, −4.66932497054279664373209399168, −3.62629167209942696266694507136, −3.10194897901318069933828856159, −1.63838559819218675215906304095, −0.35546907135486496380261916308,
0.35546907135486496380261916308, 1.63838559819218675215906304095, 3.10194897901318069933828856159, 3.62629167209942696266694507136, 4.66932497054279664373209399168, 5.06428545934688902540640276821, 5.78013002454355727943672360058, 6.71479890998734701163642680773, 7.48494528308985693665506200301, 7.83798066208240394023804992897
