| L(s) = 1 | + 3.60·2-s + 7.96·3-s + 5.01·4-s − 17.8·5-s + 28.7·6-s − 17.9·7-s − 10.7·8-s + 36.4·9-s − 64.5·10-s + 24.9·11-s + 39.9·12-s − 54.6·13-s − 64.9·14-s − 142.·15-s − 78.9·16-s − 53.5·17-s + 131.·18-s − 17.2·19-s − 89.6·20-s − 143.·21-s + 90.1·22-s − 85.8·24-s + 194.·25-s − 197.·26-s + 75.2·27-s − 90.2·28-s − 84.7·29-s + ⋯ |
| L(s) = 1 | + 1.27·2-s + 1.53·3-s + 0.626·4-s − 1.59·5-s + 1.95·6-s − 0.971·7-s − 0.476·8-s + 1.34·9-s − 2.04·10-s + 0.684·11-s + 0.960·12-s − 1.16·13-s − 1.23·14-s − 2.45·15-s − 1.23·16-s − 0.763·17-s + 1.72·18-s − 0.208·19-s − 1.00·20-s − 1.48·21-s + 0.873·22-s − 0.730·24-s + 1.55·25-s − 1.48·26-s + 0.536·27-s − 0.608·28-s − 0.542·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 \) |
| good | 2 | \( 1 - 3.60T + 8T^{2} \) |
| 3 | \( 1 - 7.96T + 27T^{2} \) |
| 5 | \( 1 + 17.8T + 125T^{2} \) |
| 7 | \( 1 + 17.9T + 343T^{2} \) |
| 11 | \( 1 - 24.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 54.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 53.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 17.2T + 6.85e3T^{2} \) |
| 29 | \( 1 + 84.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 104.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 190.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 111.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 291.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 504.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 54.5T + 1.48e5T^{2} \) |
| 59 | \( 1 - 301.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 834.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 4.71T + 3.00e5T^{2} \) |
| 71 | \( 1 - 921.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 252.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 249.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 985.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 146.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 72.7T + 9.12e5T^{2} \) |
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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
−9.711325871710003854621646298343, −9.057409597213255729800783580722, −8.140601859538590856552273639454, −7.25221634759600364665526757634, −6.42987221457323036612200113956, −4.71879015013271991225958819381, −3.98463906255838052696466709025, −3.33130416225066460371102250747, −2.50699019834661650184462914691, 0,
2.50699019834661650184462914691, 3.33130416225066460371102250747, 3.98463906255838052696466709025, 4.71879015013271991225958819381, 6.42987221457323036612200113956, 7.25221634759600364665526757634, 8.140601859538590856552273639454, 9.057409597213255729800783580722, 9.711325871710003854621646298343
