| L(s) = 1 | − 0.473·2-s + 1.68·3-s − 1.77·4-s + 1.70·5-s − 0.796·6-s − 4.37·7-s + 1.78·8-s − 0.170·9-s − 0.805·10-s − 3.42·11-s − 2.98·12-s + 4.73·13-s + 2.06·14-s + 2.86·15-s + 2.70·16-s − 5.61·17-s + 0.0805·18-s − 7.42·19-s − 3.02·20-s − 7.35·21-s + 1.61·22-s + 6.73·23-s + 3.00·24-s − 2.10·25-s − 2.23·26-s − 5.33·27-s + 7.76·28-s + ⋯ |
| L(s) = 1 | − 0.334·2-s + 0.971·3-s − 0.888·4-s + 0.761·5-s − 0.325·6-s − 1.65·7-s + 0.631·8-s − 0.0567·9-s − 0.254·10-s − 1.03·11-s − 0.862·12-s + 1.31·13-s + 0.553·14-s + 0.739·15-s + 0.676·16-s − 1.36·17-s + 0.0189·18-s − 1.70·19-s − 0.675·20-s − 1.60·21-s + 0.345·22-s + 1.40·23-s + 0.613·24-s − 0.420·25-s − 0.439·26-s − 1.02·27-s + 1.46·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9739092850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9739092850\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 97 | \( 1 \) |
| good | 2 | \( 1 + 0.473T + 2T^{2} \) |
| 3 | \( 1 - 1.68T + 3T^{2} \) |
| 5 | \( 1 - 1.70T + 5T^{2} \) |
| 7 | \( 1 + 4.37T + 7T^{2} \) |
| 11 | \( 1 + 3.42T + 11T^{2} \) |
| 13 | \( 1 - 4.73T + 13T^{2} \) |
| 17 | \( 1 + 5.61T + 17T^{2} \) |
| 19 | \( 1 + 7.42T + 19T^{2} \) |
| 23 | \( 1 - 6.73T + 23T^{2} \) |
| 29 | \( 1 + 0.543T + 29T^{2} \) |
| 31 | \( 1 - 2.63T + 31T^{2} \) |
| 37 | \( 1 + 2.67T + 37T^{2} \) |
| 41 | \( 1 - 3.99T + 41T^{2} \) |
| 43 | \( 1 - 1.57T + 43T^{2} \) |
| 47 | \( 1 + 7.32T + 47T^{2} \) |
| 53 | \( 1 - 3.73T + 53T^{2} \) |
| 59 | \( 1 + 5.67T + 59T^{2} \) |
| 61 | \( 1 - 1.20T + 61T^{2} \) |
| 67 | \( 1 - 1.38T + 67T^{2} \) |
| 71 | \( 1 + 3.57T + 71T^{2} \) |
| 73 | \( 1 + 9.44T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 + 0.828T + 89T^{2} \) |
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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
−8.009318575619778683935247954432, −7.01924596808519696588005386410, −6.28032302962580666997065128679, −5.86043400805583862499404162731, −4.86592961659290668044220602050, −4.02419000914151790415370642267, −3.35543752532492673427775962583, −2.65540612890993030743860640968, −1.89786582559845816344834600640, −0.45136980460627355157490416284,
0.45136980460627355157490416284, 1.89786582559845816344834600640, 2.65540612890993030743860640968, 3.35543752532492673427775962583, 4.02419000914151790415370642267, 4.86592961659290668044220602050, 5.86043400805583862499404162731, 6.28032302962580666997065128679, 7.01924596808519696588005386410, 8.009318575619778683935247954432
