| L(s) = 1 | − 1.59e18·2-s − 3.81e28·3-s + 1.88e36·4-s + 5.86e40·5-s + 6.09e46·6-s − 2.05e50·7-s − 1.95e54·8-s + 8.54e56·9-s − 9.36e58·10-s − 9.77e60·11-s − 7.20e64·12-s − 6.38e65·13-s + 3.27e68·14-s − 2.23e69·15-s + 1.87e72·16-s + 1.67e73·17-s − 1.36e75·18-s + 1.05e76·19-s + 1.10e77·20-s + 7.82e78·21-s + 1.56e79·22-s + 9.13e80·23-s + 7.45e82·24-s − 1.47e83·25-s + 1.02e84·26-s − 9.73e84·27-s − 3.87e86·28-s + ⋯ |
| L(s) = 1 | − 1.96·2-s − 1.55·3-s + 2.84·4-s + 0.151·5-s + 3.05·6-s − 1.06·7-s − 3.61·8-s + 1.42·9-s − 0.296·10-s − 0.106·11-s − 4.42·12-s − 0.335·13-s + 2.09·14-s − 0.235·15-s + 4.23·16-s + 1.03·17-s − 2.79·18-s + 0.862·19-s + 0.429·20-s + 1.66·21-s + 0.208·22-s + 0.866·23-s + 5.62·24-s − 0.977·25-s + 0.657·26-s − 0.663·27-s − 3.03·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(120-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+59.5) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(60)\) |
\(\approx\) |
\(0.2776114577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2776114577\) |
| \(L(\frac{121}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 1.59e18T + 6.64e35T^{2} \) |
| 3 | \( 1 + 3.81e28T + 5.99e56T^{2} \) |
| 5 | \( 1 - 5.86e40T + 1.50e83T^{2} \) |
| 7 | \( 1 + 2.05e50T + 3.68e100T^{2} \) |
| 11 | \( 1 + 9.77e60T + 8.42e123T^{2} \) |
| 13 | \( 1 + 6.38e65T + 3.62e132T^{2} \) |
| 17 | \( 1 - 1.67e73T + 2.65e146T^{2} \) |
| 19 | \( 1 - 1.05e76T + 1.48e152T^{2} \) |
| 23 | \( 1 - 9.13e80T + 1.11e162T^{2} \) |
| 29 | \( 1 + 7.45e86T + 1.06e174T^{2} \) |
| 31 | \( 1 - 8.54e88T + 2.96e177T^{2} \) |
| 37 | \( 1 + 1.06e92T + 4.13e186T^{2} \) |
| 41 | \( 1 - 7.85e95T + 8.34e191T^{2} \) |
| 43 | \( 1 - 3.24e96T + 2.41e194T^{2} \) |
| 47 | \( 1 + 2.08e99T + 9.54e198T^{2} \) |
| 53 | \( 1 - 1.04e102T + 1.54e205T^{2} \) |
| 59 | \( 1 + 2.56e105T + 5.38e210T^{2} \) |
| 61 | \( 1 + 8.19e105T + 2.84e212T^{2} \) |
| 67 | \( 1 + 5.67e108T + 2.00e217T^{2} \) |
| 71 | \( 1 - 2.10e110T + 1.99e220T^{2} \) |
| 73 | \( 1 - 3.49e110T + 5.43e221T^{2} \) |
| 79 | \( 1 + 4.77e112T + 6.57e225T^{2} \) |
| 83 | \( 1 + 2.60e114T + 2.34e228T^{2} \) |
| 89 | \( 1 - 3.17e115T + 9.49e231T^{2} \) |
| 97 | \( 1 + 1.43e118T + 2.66e236T^{2} \) |
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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
−12.37815916911026117677061409745, −11.40869225963944721748582035519, −10.20872992989062804508002375318, −9.491524421480476130901456554391, −7.62115281441976013158854065261, −6.53681117370886585150223612809, −5.64137647741168427494054058482, −2.96210493329842226584665086333, −1.29468016248445498616990665543, −0.42399222610035565811023874292,
0.42399222610035565811023874292, 1.29468016248445498616990665543, 2.96210493329842226584665086333, 5.64137647741168427494054058482, 6.53681117370886585150223612809, 7.62115281441976013158854065261, 9.491524421480476130901456554391, 10.20872992989062804508002375318, 11.40869225963944721748582035519, 12.37815916911026117677061409745
