| L(s) = 1 | − 1.05·2-s + 3-s − 0.892·4-s + 3.01·5-s − 1.05·6-s + 3.77·7-s + 3.04·8-s + 9-s − 3.17·10-s + 4.93·11-s − 0.892·12-s − 5.50·13-s − 3.97·14-s + 3.01·15-s − 1.41·16-s − 7.70·17-s − 1.05·18-s + 5.52·19-s − 2.69·20-s + 3.77·21-s − 5.19·22-s − 7.92·23-s + 3.04·24-s + 4.08·25-s + 5.79·26-s + 27-s − 3.37·28-s + ⋯ |
| L(s) = 1 | − 0.744·2-s + 0.577·3-s − 0.446·4-s + 1.34·5-s − 0.429·6-s + 1.42·7-s + 1.07·8-s + 0.333·9-s − 1.00·10-s + 1.48·11-s − 0.257·12-s − 1.52·13-s − 1.06·14-s + 0.778·15-s − 0.354·16-s − 1.86·17-s − 0.248·18-s + 1.26·19-s − 0.601·20-s + 0.824·21-s − 1.10·22-s − 1.65·23-s + 0.621·24-s + 0.816·25-s + 1.13·26-s + 0.192·27-s − 0.637·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.556826741\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.556826741\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 193 | \( 1 + T \) |
| good | 2 | \( 1 + 1.05T + 2T^{2} \) |
| 5 | \( 1 - 3.01T + 5T^{2} \) |
| 7 | \( 1 - 3.77T + 7T^{2} \) |
| 11 | \( 1 - 4.93T + 11T^{2} \) |
| 13 | \( 1 + 5.50T + 13T^{2} \) |
| 17 | \( 1 + 7.70T + 17T^{2} \) |
| 19 | \( 1 - 5.52T + 19T^{2} \) |
| 23 | \( 1 + 7.92T + 23T^{2} \) |
| 29 | \( 1 - 4.62T + 29T^{2} \) |
| 31 | \( 1 - 3.53T + 31T^{2} \) |
| 37 | \( 1 - 5.57T + 37T^{2} \) |
| 41 | \( 1 + 4.18T + 41T^{2} \) |
| 43 | \( 1 + 0.0489T + 43T^{2} \) |
| 47 | \( 1 - 8.14T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 + 2.57T + 59T^{2} \) |
| 61 | \( 1 + 6.68T + 61T^{2} \) |
| 67 | \( 1 + 6.27T + 67T^{2} \) |
| 71 | \( 1 - 0.729T + 71T^{2} \) |
| 73 | \( 1 - 3.26T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 + 9.98T + 83T^{2} \) |
| 89 | \( 1 - 3.67T + 89T^{2} \) |
| 97 | \( 1 - 0.726T + 97T^{2} \) |
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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
−10.33204061876630218174501881786, −9.575352698575457627020203405278, −9.163591669807314737756986965287, −8.262520132049915273108152869507, −7.42447162349074169783760759991, −6.30498683739148380247337047907, −4.91797996852262014640296792585, −4.33103643199734410901656649038, −2.27642503641643580988691985926, −1.46717813739665485291800655337,
1.46717813739665485291800655337, 2.27642503641643580988691985926, 4.33103643199734410901656649038, 4.91797996852262014640296792585, 6.30498683739148380247337047907, 7.42447162349074169783760759991, 8.262520132049915273108152869507, 9.163591669807314737756986965287, 9.575352698575457627020203405278, 10.33204061876630218174501881786
